3 Specht functionality

3.1 Porting notes

Porting the Specht 2.4 package to GAP 4 did not influence the algorithms but required a completely new object oriented design of the underlying data structures. In GAP 3 records were used to represent algebra objects, modules and decomposition matrices of Specht 2.4. Further functions were stored inside of such records to provide name safety.

In Hecke objects represent all the data that was named above. The overloading mechanism the former record-internal functions to be available on the toplevel. The operation selection mechanism of GAP 4 allows one to concentrate on the computation code instead of dealing with different possible argument inputs.

Since variable argument length operations are not yet supported by GAP 4, we introduced our own dispatcher facility to enable the former possibility of passing patrition arguments as sequences of integers (see 3.3).

3.1-1 Structure of Hecke

The data structure hierarchy in GAP 4 is defined through filters and their dependencies.

IsAlgebraObj is a generic filter for the objects returned by the functions Specht (3.2-1) and Schur (3.2-7). Concretely, Specht (3.2-1) returns an IsHecke object (automatically also an IsAlgebraObj object). For design reasons IsSchur is a subfilter of IsHecke. This allows to use the same functions for Schur-algebras as for Hecke-algebras with minor restrictions.

The hierarchy of module objects is more complex. On top we have the filter IsAlgebraObjModule. Its direct descendant IsHeckeModule has IsHeckeSpecht, IsHeckePIM, IsHeckeSimple, IsFockModule and IsSchurModule as subfilters. Again the last two subfilter relations have no mathematical sense but are technically comfortable. The filter IsFockModule is superfilter of IsFockSpecht, IsFockPIM, IsFockSimple and IsFockSchurModule. Analogously, IsSchurModule is superfilter of IsSchurWeyl, IsSchurPIM, IsSchurSimple and IsFockSchurModule which itself is superfilter of IsFockSchurWeyl, IsFockSchurPIM, IsFockSchurSimple. Further, there are subfilter relations between IsFockSpecht and IsHeckeSpecht etc., IsFockSchurWeyl and IsFockSpecht etc., IsFockSchurWeyl and IsSchurWeyl etc., IsSchurWeyl and IsHeckeSpecht etc. filters.

For decomposition matrices we use the filter IsDecompositionMatrix and its subfilter IsCrystalDecompositionMatrix.

3.1-2 Renamings

To keep things as backwards compatible as possible, we tried not to change names and function signatures. But for the former H.***- and H.operations.***-style functions it makes more sense to use toplevel functions (especially when the H is not explicitly needed inside of the called operation). Here is an overview of some important changes:

3.2 Specht functions

3.2-1 Specht

Returns: object belonging to the filter IsHecke (3.1-1)

Let R be a field of characteristic 0, q a non-zero element of R, and let e be the smallest positive integer such that 1+q+...+q^e-1=0 (we set e=0 if no such integer exists). The object returned by Specht(e) allows calculations in the Grothendieck rings of the Hecke algebras H of type A which are defined over R and have parameter q. Below we also describe how to consider Hecke algebras defined over fields of positive characteristic.

Specht returns an object which contains information about the the family of Hecke algebras determined by R and q. This object needs to be passed to the most of the Hecke functions as an argument.

gap> H:=Specht(5);
<Hecke algebra with e = 5>
gap> Display(last);
Specht(e=5, S(), P(), D())
gap> IsZeroCharacteristic(last);
true

There is also a method Schur (3.2-7) for doing calculations with the q-Schur algebra. See DecompositionMatrix (3.2-8), and CrystalDecompositionMatrix (3.2-9). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.2-2 Simple information access

We allow to read/store some information from/in the algebra object returned by Specht (3.2-1) using the following functions.

Returns: e.

Returns: p.

Provides writing access to Ordering that is stored in H. The ordering influences the way how decomposition matrices are printed.

Setting this global variable the user can tell Hecke where to find decomposition matrices that are not in the library and also not in the current directory. By default this variable is set to the current directory.

3.2-3 The functions MakeSpecht, MakePIM and MakeSimple

The functions MakeSpecht, MakePIM and MakeSimple return objects belonging to the filter IsAlgebraObjModule (3.1-1) which correspond to Specht modules (IsHeckeSpecht (3.1-1)), projective indecomposable modules (IsHeckePIM (3.1-1)) and simple modules (IsHeckeSimple (3.1-1)) respectively. Hecke allows manipulation of arbitrary linear combinations of these "modules", as well as a way of inducing and restricting them, "multiplying" them and converting between these three natural bases of the Grothendieck ring. Multiplication of modules corresponds to taking a tensor productand then inducing (thus giving a module for a larger Hecke algebra). Each of these three functions can be called in four different ways, as we now describe.

In the first form, μ is a partition (either a list, or a sequence of integers) and the corresponding Specht module, PIM, or simple module (respectively), is returned.

gap> H:=Specht(5);; MakePIM(H,4,3,2);; Display(last);
P(4,3,2)

Here, x is an H-module. In this form, MakeSpecht rewrites x as a linear combination of Specht modules, if possible. Similarly, MakePIM and MakeSimple rewrite x as a linear combination of PIMs and simple modules respectively. These conversions require knowledge of the relevant decomposition matrix of H; if this is not known then fail is returned (over fields of characteristic zero, all of the decomposition matrices are known via the algorithm of [LLT96]; various other decomposition matrices are included with Hecke). For example, MakeSpecht(MakePIM(H, μ)) returns ∑_ν d_ν,μ S(ν) or fail if some of these decomposition multiplicities are not known.

gap> Display( MakeSimple( MakePIM(H,4,3,2) ) );
D(5,3,1) + 2D(4,3,2) + D(2^4,1)
gap> Display( MakeSpecht( MakeSimple( MakeSpecht(H,1,1,1,1,1) ) ) );
 - S(5) + S(4,1) - S(3,1^2) + S(2,1^3)

As the last example shows, Hecke does not always behave as expected. The reason for this is that Specht modules indexed by e-singular partitions can always be written as a linear combination of Specht modules which involve only e-regular partitions. As such, it is not always clear when two elements are equal in the Grothendieck ring. Consequently, to test whether two modules are equal you should first rewrite both modules in the D-basis; this is not done by Hecke because it would be very inefficient.

In the third form, d is a decomposition matrix and μ is a partition. This is useful when you are trying to calculate a new decomposition matrix d because it allows you to do calculations using the known entries of d to deduce information about the unknown ones. When used in this way, MakePIM and MakeSimple use d to rewrite P(μ) and D(μ) respectively as a linear combination of Specht modules and MakeSpecht uses d to write S(μ) as a linear combination of simple modules. If the values of the unknown entries in d are needed, fail is returned.

gap> H:=Specht(3,3);;   # e = 3, p = 3 = characteristic of 'R'
gap>  d:=InducedDecompositionMatrix(DecompositionMatrix(H,14));;
# Inducing....
The following projectives are missing from <d>:
    [ 15 ]  [ 8, 7 ]
gap> Display(MakePIM(d,4,3,3,2,2,1));
S(4,3^2,2^2,1) + S(4,3^2,2,1^3) + S(4,3,2^3,1^2) + S(3^3,2^2,1^2)
gap> Display(MakeSpecht(d,7, 3, 3, 2));
D(11,2,1^2) + D(10,3,1^2) + D(8,5,1^2) + D(8,3^2,1) + D(7,6,1^2) + D(7,3^2,2)
gap> Display(MakeSimple(d,14,1));
fail

The final example returned fail because the partitions (14,1) and (15) have the same 3-core (and P(15) is missing from d).

In the final form, d is a decomposition matrix and x is a module. All three functions rewrite x in their respective basis using d. Again this is only useful when you are trying to calculate a new decomposition matrix because, for any "known" decomposition matrix d, MakeSpecht(x) and MakeSpecht(d, x) are equivalent (and similarly for MakePIM and MakeSimple).

gap> Display(MakeSpecht(d, MakeSimple(d,10,5) ));
 - S(13,2) + S(10,5)

3.2-4 Decomposition numbers of the symmetric groups

The last example looked at Hecke algebras with parameter q=1 and R a field of characteristic 3 (so e=3); that is, the group algebra of the symmetric group over a field of characteristic 3. More generally, the command Specht(p,p) can be used to consider the group algebras of the symmetric groups over fields of characteristic p (i.e. e=p and R a field of characteristic p). For example, the dimensions of the simple modules of S_6 over fields of characteristic 5 can be computed as follows:

gap> H:=Specht(5,5);; SimpleDimension(H,6);
6       : 1
5,1     : 5
4,2     : 8
4,1^2   : 10
3^2     : 5
3,2,1   : 8
3,1^3   : 10
2^3     : 5
2^2,1^2 : 1
2,1^4   : 5
true

3.2-5 Hecke algebras over fields of positive characteristic

To consider Hecke algebras defined over arbitrary fields, Specht (3.2-1) must also be supplied with a valuation map val as an argument. The function val is a map from some PID into the natural numbers; at present it is needed only by functions which rely (at least implicitly), upon the q-analogue of Schaper's theorem. In general, val depends upon q and the characteristic of R; full details can be found in [JM97]. Over fields of characteristic zero and in the symmetric group case, the function val is automatically defined by Specht (3.2-1). When R is a field of characteristic zero, val([i]_q) is 1 if e divides i and 0 otherwise (this is the valuation map associated to the prime ideal in ℂ[v] generated by the e-th cyclotomic polynomial). When q=1 and R is a field of characteristic p, val is the usual p-adic valuation map. As another example, if q=4 and R is a field of characteristic 5 (so e=2), then the valuation map sends the integer x to ν_5([4]_x) where [4]_x is interpreted as an integer and ν_5 is the usual 5-adic valuation. To consider this Hecke algebra one could proceed as follows:

gap> val:=function(x) local v;
>       x:=Sum([0..x-1],v->4^v);  # x->[x]_q
>       v:=0; while x mod 5=0 do x:=x/5; v:=v+1; od;
>       return v;
>     end;;
gap> H:=Specht(2,5,val,"e2q4");; Display(H);
Specht(e=2, p=5, S(), P(), D(), HeckeRing="e2q4")

Notice the string "e2q4" which was also passed to Specht (3.2-1) in this example. Although it is not strictly necessary, it is a good idea when using a "non-standard" valuation map val to specify the value of HeckeRing. This string is used for internal bookkeeping by Hecke; in particular, it is used to determine filenames when reading and saving decomposition matrices. If a "standard" valuation map is used then HeckeRing is set to the string "e<e>p<p>"; otherwise it defaults to "unknown". The function SaveDecompositionMatrix (3.5-5) will not save any decomposition matrix for any Hecke algebra H with HeckeRing="unknown".

3.2-6 The Fock space and Hecke algebras over fields of characteristic zero

For Hecke algebras H defined over fields of characteristic zero Lascoux, Leclerc and Thibon [LLT96] have described an easy, inductive, algorithm for calculating the decomposition matrices of H. Their algorithm really calculates the canonical basis, or (global) crystal basis of the Fock space; results of Grojnowski-Lusztig [Gro94] show that computing this basis is equivalent to computing the decomposition matrices of H (see also [Ari96]).

The Fock space F is an (integrable) module for the quantum group U_q( sl^_e) of the affine special linear group. F is a free ℂ[v]-module with basis the set of all Specht modules S(μ) for all partitions μ of all integers.

Here v is an indeterminate over the integers (or strictly, ℂ) that is stored in the algebra object produced by Specht (3.2-1). The canonical basis elements Pq(μ) for the U_q( sl^_e)-submodule of F generated by the 0-partition are indexed by e-regular partitions μ. Moreover, under specialization, Pq(μ) maps to P(μ). An eloquent description of the algorithm for computing Pq(μ) can be found in [LLT96].

To access the elements of the Fock space Hecke provides the functions:

Notice that, unlike MakePIM (3.2-3) and MakeSpecht (3.2-3), the only arguments which MakeFockPIM and MakeFockSpecht accept are partitions.

The function MakeFockPIM computes the canonical basis element Pq(μ) of the Fock space corresponding to the e-regular partition μ (there is a canonical basis -- defined using a larger quantum group -- for the whole of the Fock space [LT96]; conjecturally, this basis can be used to compute the decomposition matrices for the q-Schur algebra over fields of characteristic zero). The second function returns a standard basis element Sq(μ) of F.

gap> H:=Specht(4);; MakeFockPIM(H,6,2);; Display(last);
Sq(6,2) + vSq(5,3)
gap> RRestrictedModule(last); Display(last);
Sq(6,1) + (v+v^-1)Sq(5,2) + vSq(4,3)
gap> MakePIM(last);; Display(last);
Pq(6,1) + (v+v^-1)Pq(5,2)
gap> Specialized(last);; Display(last);
P(6,1) + 2P(5,2)
gap> MakeFockSpecht(H,5,3,2);; Display(last);
Sq(5,3,2)
gap> RInducedModule(last,0);; Display(last);
v^-1Sq(5,3^2)

The modules returned by MakeFockPIM and MakeFockSpecht behave very much like elements of the Grothendieck ring of H; however, they should be considered as elements of the Fock space. The key difference is that when induced or restricted "quantum" analogues of induction and restriction are used. These analogues correspond to the action of U_q( sl^_e) on F [LLT96].

In effect, the functions MakeFockPIM and MakeFockSpecht allow computations in the Fock space, using the functions RInducedModule (3.4-1) and RRestrictedModule (3.4-3). The functions MakeSpecht (3.2-3), MakePIM (3.2-3) and MakeSimple (3.2-3) can also be applied to elements of the Fock space, in which case they have the expected effect. In addition, any element of the Fock space can be specialized to give the corresponding element of the Grothendieck ring of H (it is because of this correspondence that we do not make a distinction between elements of the Fock space and the Grothendieck ring of H).

When working over fields of characteristic zero Hecke will automatically calculate any canonical basis elements that it needs for computations in the Grothendieck ring of H. If you are not interested in the canonical basis elements you need never work with them directly.

3.2-7 Schur

Returns: object belonging to the filter IsSchur (3.1-1)

This function behaves almost identically to the function Specht (3.2-1), the only difference being that the belonging modules are printed as "W", "P", "F" and that they correspond to the q-Weyl modules, the projective indecomposable modules, and the simple modules of the q-Schur algebra respectively. Note that the Make***-functions (i.e. MakeSpecht (3.2-3) is used to generate q-Weyl modules). Further, note that our labeling of these modules is non-standard, following that used by James in [Jam90]. The standard labeling can be obtained from ours by replacing all partitions by their conjugates.

Almost all of the functions in Hecke which accept a Hecke algebra object H will also accept the object S returned by Schur .

In the current version of Hecke the decomposition matrices of q -Schur algebras are not fully supported. The InducedDecompositionMatrix (3.5-1) function can be applied to these matrices; however there are no additional routines available for calculating the columns corresponding to e-singular partitions. The decomposition matrices for the q-Schur algebras defined over a field of characteristic 0 for n≤ 10 are in the Hecke libraries.

gap> S:=Schur(2);
<Schur algebra with e = 2>
gap> InducedDecompositionMatrix(DecompositionMatrix(S,3));
# The following projectives are missing from <d>:
#  [ 2, 2 ]
<5x5 decomposition matrix>
gap> Display(last);
4    | 1                   # DecompositionMatrix(S,4) returns the
3,1  | 1 1                 # full decomposition matrix. The point
2^2  | . 1 .               # of this example is to emphasize the
2,1^2| 1 1 . 1             # current limitations of Schur.
1^4  | 1 . . 1 1

Note that when S is defined over a field of characteristic zero then the functions MakeFockSpecht (3.2-6) and MakeFockPIM (3.2-6) will calculate the canonical basis elements (see Specht (3.2-1)); currently MakeFockPIM(μ) is implemented only for e-regular partitions.

See also Specht (3.2-1). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.2-8 DecompositionMatrix

Returns: the decomposition matrix D of H(S_n) where H is a Hecke algebra object returned by the function Specht (3.2-1) (or Schur (3.2-7) ).

DecompositionMatrix first checks whether the required decomposition matrix exists as a library file (checking first in the current directory, next in the directory specified by SpechtDirectory (3.2-2), and finally in the Hecke libraries). If the base field of H has characteristic zero, DecompositionMatrix next looks for crystallized decomposition matrices (see CrystalDecompositionMatrix (3.2-9)). If the decomposition matrix d is not stored in the library DecompositionMatrix will calculate d when H is a Hecke algebra with a base field of characteristic zero, and will return fail otherwise (in which case the function CalculateDecompositionMatrix (3.5-6) can be used to force Hecke to try and calculate this matrix).

For Hecke algebras defined over fields of characteristic zero, Hecke uses the algorithm of [LLT96] to calculate decomposition matrices The decomposition matrices for the q-Schur algebras for n≤10 are contained in the Hecke library, as are those for the symmetric group over fields of positive characteristic when n<15.

Once a decomposition matrix is known, Hecke keeps an internal copy of it which is used by the functions MakeSpecht (3.2-3), MakePIM (3.2-3), and MakeSimple (3.2-3); these functions also read decomposition matrix files as needed.

If you set the variable SpechtDirectory (3.2-2), then Hecke will also search for decomposition matrix files in this directory. The files in the current directory override those in SpechtDirectory (3.2-2) and those in the Hecke libraries.

In the second form of the function, when a filename is supplied, DecompositionMatrix will read the decomposition matrix in the file filename, and this matrix will become Hecke's internal copy of this matrix.

By default, the rows and columns of the decomposition matrices are ordered DecompositionMatrix with an ordering function such as LengthLexicographic (3.8-13) or ReverseDominance (3.8-15). You do not need to specify the ordering you want every time you call DecompositionMatrix; Hecke will keep the same ordering until you change it again. This ordering can also be set "by hand" using the operation SetOrdering (3.2-2)

gap> DecompositionMatrix(Specht(3),6,LengthLexicographic);
<11x7 decomposition matrix>
gap> Display(last);
6      | 1
5,1    | 1 1
4,2    | . . 1
3^2    | . 1 . 1
4,1^2  | . 1 . . 1
3,2,1  | 1 1 . 1 1 1
2^3    | 1 . . . . 1
3,1^3  | . . . . 1 1
2^2,1^2| . . . . . . 1
2,1^4  | . . . 1 . 1 .
1^6    | . . . 1 . . .

Once you have a decomposition matrix it is often nice to be able to print it. The on screen version is often good enough; There are also functions for converting Hecke decomposition matrices into GAP matrices and vice versa (see MatrixDecompositionMatrix (3.5-7) and DecompositionMatrixMatrix (3.5-8)).

Using the function InducedDecompositionMatrix (3.5-1), it is possible to induce a decomposition matrix. See also SaveDecompositionMatrix (3.5-5) and IsNewIndecomposable (3.5-2), Specht (3.2-1), Schur (3.2-7), and CrystalDecompositionMatrix (3.2-9). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.2-9 CrystalDecompositionMatrix

Returns: the crystal decomposition matrix D of H(S_n) where H is a Hecke algebra object returned by the function Specht (3.2-1) (or Schur (3.2-7) ).

This function is similar to DecompositionMatrix (3.2-8). The columns of decomposition matrices correspond to projective indecomposables; the columns of crystallized decomposition matrices correspond to the canonical basis elements of the Fock space (see Specht (3.2-1)). Consequently, the entries in these matrices are polynomials (in v), and by specializing (i.e. setting v equal to 1; see Specialized (3.9-1)), the decomposition matrices of H are obtained (see Specht (3.2-1)). Crystallized decomposition matrices are defined only for Hecke algebras over a base field of characteristic zero. Unlike "normal" decomposition matrices, crystallized decomposition matrices cannot be induced.

gap> CrystalDecompositionMatrix(Specht(3), 6);
<11x7 decomposition matrix>
gap> Display(last);
6      |   1
5,1    |   v   1
4,2    |   .   .   1
4,1^2  |   .   v   .   1
3^2    |   .   v   .   .   1
3,2,1  |   v v^2   .   v   v   1
3,1^3  |   .   .   . v^2   .   v
2^3    | v^2   .   .   .   .   v
2^2,1^2|   .   .   .   .   .   .   1
2,1^4  |   .   .   .   .   v v^2   .
1^6    |   .   .   .   . v^2   .   .
gap> Specialized(last);   # set v equal to 1.
<11x7 decomposition matrix>
gap> Display(last);
6      | 1
5,1    | 1 1
4,2    | . . 1
4,1^2  | . 1 . 1
3^2    | . 1 . . 1
3,2,1  | 1 1 . 1 1 1
3,1^3  | . . . 1 . 1
2^3    | 1 . . . . 1
2^2,1^2| . . . . . . 1
2,1^4  | . . . . 1 1 .
1^6    | . . . . 1 . .

See also Specht (3.2-1), Schur (3.2-7), DecompositionMatrix (3.2-8) and Specialized (3.9-1). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.2-10 DecompositionNumber

This function attempts to calculate the decomposition multiplicity of D(ν) in S(μ) (equivalently, the multiplicity of S(μ) in P(ν)). If P(ν) is known, we just look up the answer; if not DecompositionNumber tries to calculate the answer using "row and column removal" (see [Jam90, Theorem 6.18]).

gap> H:=Specht(6);; DecompositionNumber(H,[6,4,2],[6,6]);
0

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.3 Partitions in Hecke

Many of the functions in Hecke take partitions as arguments. Partitions are usually represented by lists in GAP. In Hecke, all the functions which expect a partition will accept their argument either as a list or simply as a sequence of numbers. So, for example:

gap> H:=Specht(4);; Print(MakeSpecht(MakePIM(H,6,4)),"\n");
S(6,4) + S(6,3,1) + S(5,3,1,1) + S(3,3,2,1,1) + S(2,2,2,2,2)
gap> Print(MakeSpecht(MakePIM(H,[6,4])),"\n");
S(6,4) + S(6,3,1) + S(5,3,1,1) + S(3,3,2,1,1) + S(2,2,2,2,2)

Some functions require more than one argument, but the convention still applies.

gap> ECore(3, [6,4,2]);
[ 6, 4, 2 ]
gap> ECore(3, 6,4,2);
[ 6, 4, 2 ]
gap> GoodNodes(3, 6,4,2);
[ false, false, 3 ]
gap> GoodNodes(3, [6,4,2], 2);
3

Basically, it never hurts to put the extra brackets in, and they can be omitted so long as this is not ambiguous. One function where the brackets are needed is DecompositionNumber (3.2-10) this is clear because the function takes two partitions as its arguments.

3.4 Inducing and restricting modules

Hecke provides four functions RInducedModule (3.4-1), RRestrictedModule (3.4-3), SInducedModule (3.4-2) and SRestrictedModule (3.4-4) for inducing and restricting modules. All functions can be applied to Specht modules, PIMs, and simple modules. These functions all work by first rewriting all modules as a linear combination of Specht modules (or q-Weyl modules), and then inducing and restricting. Whenever possible the induced or restricted module will be written in the original basis.

All of these functions can also be applied to elements of the Fock space (see Specht (3.2-1)); in which case they correspond to the action of the generators E_i and F_i of U_q( sl^_e) on F. There is also a function InducedDecompositionMatrix (3.5-1) for inducing decomposition matrices.

3.4-1 RInducedModule

Returns: the induced modules of the Specht modules, principal indecomposable modules, and simple modules (more accurately, their image in the Grothendieck ring).

There is an natural embedding of H(S_n) in H(S_n+1) which in the usual way lets us define an induced H(S_n+1)-module for every H(S_n)-module.

There is also a function SInducedModule (3.4-2) which provides a much faster way of r-inducing s times (and inducing s times).

Let μ be a partition. Then the induced module RInducedModule(S(μ)) is easy to describe: it has the same composition factors as ∑ S(ν) where ν runs over all partitions whose diagrams can be obtained by adding a single node to the diagram of μ.

gap> H:=Specht(2,2);;
gap> Display(RInducedModule(MakeSpecht(H,7,4,3,1)));
S(8,4,3,1) + S(7,5,3,1) + S(7,4^2,1) + S(7,4,3,2) + S(7,4,3,1^2)
gap> Display(RInducedModule(MakePIM(H,5,3,1)));
P(6,3,1) + 2P(5,4,1) + P(5,3,2)
gap> Display(RInducedModule(MakeSimple(H,11,2,1)));
# D(<x>), unable to rewrite <x> as a sum of simples
S(12,2,1) + S(11,3,1) + S(11,2^2) + S(11,2,1^2)

When inducing indecomposable modules and simple modules, RInducedModule first rewrites these modules as a linear combination of Specht modules (using known decomposition matrices), and then induces this linear combination of Specht modules. If possible Hecke then rewrites the induced module back in the original basis. Note that in the last example above, the decomposition matrix for S_15 is not known by Hecke this is why RInducedModule was unable to rewrite this module in the D-basis.

r-Induction

Two Specht modules S(μ) and S(ν) belong to the same block if and only if the corresponding partitions μ and ν have the same e-core [JM97] (see ECore (3.8-1)). Because the e-core of a partition is determined by its (multiset of) e -residues, if S(μ) and S(ν) appear in RInducedModule(S(τ)), for some partition τ, then S(μ) and S(nu) belong to the same block if and only if μ and ν can be obtained by adding a node of the same e-residue to the diagram of τ. The second form of RInducedModule allows one to induce "within blocks" by only adding nodes of some fixed e-residue r; this is known as r-induction. Note that 0≤ r<e.

gap> H:=Specht(4);; Display(RInducedModule(MakeSpecht(H,5,2,1)));
S(6,2,1) + S(5,3,1) + S(5,2^2) + S(5,2,1^2)
gap> Display(RInducedModule(MakeSpecht(H,5,2,1),0));
0S()
gap> Display(RInducedModule(MakeSpecht(H,5,2,1),1));
S(6,2,1) + S(5,3,1) + S(5,2,1^2)
gap> Display(RInducedModule(MakeSpecht(H,5,2,1),2));
0S()
gap> Display(RInducedModule(MakeSpecht(H,5,2,1),3));
S(5,2^2)

The function EResidueDiagram (3.7-13), prints the diagram of μ, labeling each node with its e-residue. A quick check of this diagram confirms the answers above.

gap> EResidueDiagram(H,5,2,1);
  0  1  2  3  0
  3  0
  2
true

"Quantized" induction

When RInducedModule is applied to the canonical basis elements MakeFockPIM(μ) (or more generally elements of the Fock space; see Specht (3.2-1)), a "quantum analogue" of induction is applied. More precisely, the function RInducedModule(*,i) corresponds to the action of the generator F_i of the quantum group U_q( sl^_e) on F [LLT96].

gap> H:=Specht(3);; x:=RInducedModule(MakeFockPIM(H,4,2),1,2);;
gap> Display(x); Display(MakePIM(x));
Sq(6,2) + vSq(4^2) + v^2Sq(4,2^2)
Pq(6,2)

See also SInducedModule (3.4-2), RRestrictedModule (3.4-3) and SRestrictedModule (3.4-4). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.4-2 SInducedModule

The function SInducedModule, standing for "string induction", provides a more efficient way of r-inducing s times (and a way of inducing s times if the residue r is omitted); r-induction is explained in "RInducedModule (3.4-1).

gap> SizeScreen([80,20]);;
gap> H:=Specht(4);; Display(SInducedModule(MakePIM(H,5,2,1),3));
P(8,2,1) + 3P(7,3,1) + 2P(7,2^2) + 6P(6,3,2) + 6P(6,3,1^2) + 3P(6,2,1^3) + 2P(\
5,3^2) + P(5,2^2,1^2)
gap> Display(SInducedModule(MakePIM(H,5,2,1),3,1));
P(6,3,1^2)
gap> Display(RInducedModule(MakePIM(H,5,2,1),1,1,1));
6P(6,3,1^2)

Note that the multiplicity of each summand of RInducedModule(x,r,...,r) is divisible by s! and that SInducedModule divides by this constant.

As with RInducedModule (3.4-1) this function can also be applied to elements of the Fock space (see Specht (3.2-1)), in which case the quantum analogue of induction is used.

See also RInducedModule (3.4-1). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.4-3 RRestrictedModule

Returns: the corresponding module for H(S_n-1) when given a module x for H(S_n)

The restriction of the Specht module S(μ) is the linear combination of Specht modules ∑ S(ν) where ν runs over the partitions whose diagrams are obtained by deleting a node from the diagram of μ. If only nodes of residue r are deleted then this corresponds to first restricting S(μ) and then taking one of the block components of the restriction; this process is known as r-restriction (cf. r-induction in RInducedModule (3.4-1)).

There is also a function SRestrictedModule (3.4-4) which provides a faster way of r-restricting s times (and restricting s times).

When more than one residue if given to RRestrictedModule it returns RRestrictedModule(x,r_1,r_2,...,r_k)= RRestrictedModule( RRestrictedModule(x,r_1),r_2,...,r_k) (cf. RInducedModule (3.4-1)).

gap> H:=Specht(6);; Display(RRestrictedModule(MakePIM(H,5,3,2,1),4));
2P(4,3,2,1)
gap> Display(RRestrictedModule(MakeSimple(H,5,3,2),1));
D(5,2^2)

"Quantized" restriction

As with RInducedModule (3.4-1), when RRestrictedModule is applied to the canonical basis elements MakeFockPIM(μ) a quantum analogue of restriction is applied; this time, RRestrictedModule(*,i) corresponds to the action of the generator E_i of U_q( sl^_e) on F[LLT96].

See also RInducedModule (3.4-1), SInducedModule (3.4-2) and SRestrictedModule (3.4-4). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.4-4 SRestrictedModule

As with SInducedModule (3.4-2) this function provides a more efficient way of r-restricting s times, or restricting s times if the residue r is omitted (cf. SInducedModule (3.4-2)).

gap> H:=Specht(6);; Display(SRestrictedModule(MakeSpecht(H,4,3,2),3));
3S(4,2) + 2S(4,1^2) + 3S(3^2) + 6S(3,2,1) + 2S(2^3)
gap> Display(SRestrictedModule(MakePIM(H,5,4,1),2,4));
P(4^2)

See also RInducedModule (3.4-1), SInducedModule (3.4-2) and RRestrictedModule (3.4-3). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.5 Operations on decomposition matrices

Hecke is a package for computing decomposition matrices; this section describes the functions available for accessing these matrices directly. In addition to decomposition matrices, Hecke also calculates the "crystallized decomposition matrices" of [LLT96] and the "adjustment matrices" introduced by James [Jam90] (and Geck [Gec92]).

Throughout Hecke we place an emphasis on calculating the projective indecomposable modulesand hence upon the columns of decomposition matrices. This approach seems more efficient than the traditional approach of calculating decomposition matrices by rows; ideally both approaches should be combined (as is done by IsNewIndecomposable (3.5-2)).

In principle, all decomposition matrices for all Hecke algebras defined over a field of characteristic zero are available from within Hecke. In addition, the decomposition matrices for all q-Schur algebras with n≤10 and all values of e and the p-modular decomposition matrices of the symmetric groups S_n for n<15 are in the Hecke library files.

If you are using Hecke regularly to do calculations involving certain values of e it would be advantageous to have Hecke calculate and save the first 20 odd decomposition matrices that you are interested in. So, for e=4 use the commands:

gap> H:=Specht(4);; for n in [8..20] do
>      SaveDecompositionMatrix(DecompositionMatrix(H,n));
>    od;

Alternatively, you could save the crystallized decomposition matrices. Note that for n<2e the decomposition matrices are known (by Hecke) and easy to compute.

3.5-1 InducedDecompositionMatrix

If d is the decomposition matrix of H(S_n), then InducedDecompositionMatrix(d) attempts to calculate the decomposition matrix of H(S_n+1). It does this by extracting each projective indecomposable from d and inducing these modules to obtain projective modules for H(S_n+1). InducedDecompositionMatrix then tries to decompose these projectives using the function IsNewIndecomposable (3.5-2). In general there will be columns of the decomposition matrix which InducedDecompositionMatrix is unable to decompose and these will have to be calculated "by hand". InducedDecompositionMatrix prints a list of those columns of the decomposition matrix which it is unable to calculate (this list is also printed by the function MissingIndecomposables (3.5-11)).

gap> d:=DecompositionMatrix(Specht(3,3),14);
<135x57 decomposition matrix>
gap> InducedDecompositionMatrix(d);
# Inducing....
The following projectives are missing from <d>:
    [ 15 ]  [ 8, 7 ]
<176x70 decomposition matrix>

Note that the missing indecomposables come in "pairs" which map to each other under the Mullineux map (see MullineuxMap (3.7-3)).

Almost all of the decomposition matrices included in Hecke were calculated directly by InducedDecompositionMatrix. When n is "small" InducedDecompositionMatrix is usually able to return the full decomposition matrix for H(S_n). Finally, although the InducedDecompositionMatrix can also be applied to the decomposition matrices of the q-Schur algebras (see Schur (3.2-7)), InducedDecompositionMatrix is much less successful in inducing these decomposition matrices because it contains no special routines for dealing with the indecomposable modules of the q-Schur algebra which are indexed by e-singular partitions. Note also that we use a non-standard labeling of the decomposition matrices of q-Schur algebras; see Schur (3.2-7).

3.5-2 IsNewIndecomposable

Returns: true if it is able to show that x is indecomposable (and this indecomposable is not already listed in d), and false otherwise.

IsNewIndecomposable is the function which does all of the hard work when the function InducedDecompositionMatrix (3.5-1) is applied to decomposition matrices. IsNewIndecomposable will also print a brief description of its findings, giving an upper and lower bound on the first decomposition number μ for which it is unable to determine the multiplicity of S(μ) in x.

IsNewIndecomposable works by running through all of the partitions ν such that P(ν) could be a summand of x and it uses various results, such as the q-Schaper theorem of [JM97] (see Schaper (3.7-1)), the Mullineux map (see MullineuxMap (3.7-3)) and inducing simple modules, to determine if P(ν) does indeed split off. In addition, if d is the decomposition matrix for H(S_n) then IsNewIndecomposable will probably use some of the decomposition matrices of H(S_m) for m≤ n, if they are known. Consequently it is a good idea to save decomposition matrices as they are calculated (see SaveDecompositionMatrix (3.5-5)).

For example, in calculating the 2-modular decomposition matrices of S_r the first projective which InducedDecompositionMatrix (3.5-1) is unable to calculate is P(10).

gap> H:=Specht(2,2);;
gap> d:=InducedDecompositionMatrix(DecompositionMatrix(H,9));;
# Inducing.
# The following projectives are missing from <d>:
#  [ 10 ]

(In fact, given the above commands, Hecke will return the full decomposition matrix for S_10 because this matrix is in the library; these were the commands that were used to calculate the decomposition matrix in the library.)

By inducing P(9) we can find a projective H-module which contains P(10). We can then use IsNewIndecomposable to try and decompose this induced module into a sum of PIMs.

gap> SizeScreen([80,20]);; x:=RInducedModule(MakePIM(H,9),1);; Display(x);
# P(<x>), unable to rewrite <x> as a sum of projectives
S(10) + S(9,1) + S(8,2) + 2S(8,1^2) + S(7,3) + 2S(7,1^3) + 3S(6,3,1) + 3S(6,2^\
2) + 4S(6,2,1^2) + 2S(6,1^4) + 4S(5,3,2) + 5S(5,3,1^2) + 5S(5,2^2,1) + 2S(5,1^\
5) + 2S(4^2,2) + 2S(4^2,1^2) + 2S(4,3^2) + 5S(4,3,1^3) + 2S(4,2^3) + 5S(4,2^2,\
1^2) + 4S(4,2,1^4) + 2S(4,1^6) + 2S(3^3,1) + 2S(3^2,2^2) + 4S(3^2,2,1^2) + 3S(\
3^2,1^4) + 3S(3,2^2,1^3) + 2S(3,1^7) + S(2^3,1^4) + S(2^2,1^6) + S(2,1^8) + S(\
1^10)
gap> IsNewIndecomposable(d,x);
# The multiplicity of S(6,3,1) in P(10) is at least 1 and at most 2.
false
gap> Display(x);
S(10) + S(9,1) + S(8,2) + 2S(8,1^2) + S(7,3) + 2S(7,1^3) + 2S(6,3,1) + 2S(6,2^\
2) + 3S(6,2,1^2) + 2S(6,1^4) + 3S(5,3,2) + 4S(5,3,1^2) + 4S(5,2^2,1) + 2S(5,1^\
5) + 2S(4^2,2) + 2S(4^2,1^2) + 2S(4,3^2) + 4S(4,3,1^3) + 2S(4,2^3) + 4S(4,2^2,\
1^2) + 3S(4,2,1^4) + 2S(4,1^6) + 2S(3^3,1) + 2S(3^2,2^2) + 3S(3^2,2,1^2) + 2S(\
3^2,1^4) + 2S(3,2^2,1^3) + 2S(3,1^7) + S(2^3,1^4) + S(2^2,1^6) + S(2,1^8) + S(\
1^10)

Notice that some of the coefficients of the Specht modules in x have changed; this is because IsNewIndecomposable was able to determine that the multiplicity of S(6,3,1) was at most 2 and so it subtracted one copy of P(6,3,1) from x.

In this case, the multiplicity of S(6,3,1) in P(10) is easy to resolve because general theory says that this multiplicity must be odd. Therefore, x-P(6,3,1) is projective. After subtracting P(6,3,1) from x we again use IsNewIndecomposable to see if x is now indecomposable. We can tell IsNewIndecomposable that all of the multiplicities up to and including S(6,3,1) have already been checked by giving it the addition argument μ=[6,3,1].

gap> x:=x-MakePIM(d,6,3,1);; IsNewIndecomposable(d,x,6,3,1);
true

Consequently, x=P(10) and we add it to the decomposition matrix d (and save it).

gap> AddIndecomposable(d,x); SaveDecompositionMatrix(d);

A full description of what IsNewIndecomposable does can be found by reading the comments in specht.gi. Any suggestions or improvements on this function would be especially welcome.

See also DecompositionMatrix (3.2-8) and InducedDecompositionMatrix (3.5-1). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.5-3 InvertDecompositionMatrix

Returns: inverse of the (e-regular part of) d, where d is a decomposition matrix, or crystallized decomposition matrix, of a Hecke algebra or q-Schur algebra.

If part of the decomposition matrix d is unknown then InvertDecompositionMatrix will invert as much of d as possible.

gap> H:=Specht(4);; d:=CrystalDecompositionMatrix(H,5);;
gap> Display(InvertDecompositionMatrix(d));
5    |   1
4,1  |   .   1
3,2  |  -v   .   1
3,1^2|   .   .   .   1
2^2,1| v^2   .  -v   .   1
2,1^3|   .   .   .   .   .   1

See also DecompositionMatrix (3.2-8) and CrystalDecompositionMatrix (3.2-9). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.5-4 AdjustmentMatrix

Returns: the adjustment matrix a

James [Jam90] noticed and Geck [Gec92] proved, that the decomposition matrices dp for Hecke algebras defined over fields of positive characteristic admit a factorization dp = d ⋅ a where d is a decomposition matrix for a suitable Hecke algebra defined over a field of characteristic zero and a is the so-called adjustment matrix.

gap> H:=Specht(2);; Hp:=Specht(2,2);;
gap> d:=DecompositionMatrix(H,13);; dp:=DecompositionMatrix(Hp,13);;
gap> a:=AdjustmentMatrix(dp,d);
<18x18 decomposition matrix>
gap> Display(a);
13     | 1
12,1   | . 1
11,2   | 1 . 1
10,3   | . . . 1
10,2,1 | . . . . 1
9,4    | 1 . 1 . . 1
9,3,1  | 2 . . . . . 1
8,5    | . 1 . . . . . 1
8,4,1  | 1 . . . . . . . 1
8,3,2  | . 2 . . . . . 1 . 1
7,6    | 1 . . . . 1 . . . . 1
7,5,1  | . . . . . . 1 . . . . 1
7,4,2  | 1 . 1 . . 1 . . . . 1 . 1
7,3,2,1| . . . . . . . . . . . . . 1
6,5,2  | . 1 . . . . . 1 . 1 . . . . 1
6,4,3  | 2 . . . 1 . . . . . . . . . . 1
6,4,2,1| . 2 . 1 . . . . . . . . . . . . 1
5,4,3,1| 4 . 2 . . . . . . . . . . . . . . 1
gap> MatrixDecompositionMatrix(dp)=
>           MatrixDecompositionMatrix(d)*MatrixDecompositionMatrix(a);
true

In the last line we have checked our calculation.

See also DecompositionMatrix (3.2-8) and CrystalDecompositionMatrix (3.2-9). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.5-5 SaveDecompositionMatrix

The function SaveDecompositionMatrix saves the decomposition matrix d. After a decomposition matrix has been saved, the functions MakeSpecht (3.2-3), MakePIM (3.2-3) and MakeSimple (3.2-3) will automatically access it as needed. So, for example, before saving d in order to retrieve the indecomposable P(μ) from d it is necessary to type MakePIM(d, μ); once d has been saved, the command MakePIM(μ) suffices.

Since InducedDecompositionMatrix (3.5-1) consults the decomposition matrices for smaller n, if they are available, it is advantageous to save decomposition matrices as they are calculated. For example, over a field of characteristic 5, the decomposition matrices for the symmetric groups S_n with n≤ 20 can be calculated as follows:

gap> H:=Specht(5,5);;
gap> d:=DecompositionMatrix(H,9);;
gap> for r in [10..20] do
>      d:=InducedDecompositionMatrix(d);
>      SaveDecompositionMatrix(d);
>    od;
# Inducing...
# Inducing....
# Inducing....
# Inducing.....
# Inducing......
# Inducing.......
# Inducing........
# Inducing..........
# Inducing............
# Inducing..............
# Inducing.................

If your Hecke algebra object H is defined using a non-standard valuation map (see Specht (3.2-1)) then it is also necessary to set the string HeckeRing, or to supply the function with a filename before it will save your matrix. SaveDecompositionMatrix will also save adjustment matrices and the various other matrices that appear in Hecke (they can be read back in using DecompositionMatrix (3.2-8)). Each matrix has a default filename which you can over ride by supplying a filename. Using non-standard file names will stop Hecke from automatically accessing these matrices in future. See also DecompositionMatrix (3.2-8) and CrystalDecompositionMatrix (3.2-9). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.5-6 CalculateDecompositionMatrix

CalculateDecompositionMatrix is similar to the function DecompositionMatrix (3.2-8) in that both functions try to return the decomposition matrix d of H(S_n); the difference is that this function tries to calculate this matrix whereas the latter reads the matrix from the library files (in characteristic zero both functions apply the algorithm of [LLT96] to compute d). In effect this function is only needed when working with Hecke algebras defined over fields of positive characteristic (or when you wish to avoid the libraries). For example, if you want to do calculations with the decomposition matrix of the symmetric group S_15 over a field of characteristic two, DecompositionMatrix (3.2-8) returns fail whereas CalculateDecompositionMatrix returns a part of the decomposition matrix.

gap> H:=Specht(2,2);; d:=DecompositionMatrix(H,15);
# This decomposition matrix is not known; use CalculateDecompositionMatrix()
# or InducedDecompositionMatrix() to calculate with this matrix.
fail
gap> d:=CalculateDecompositionMatrix(H,15);;
# Projective indecomposable P(6,4,3,2) not known.
# Projective indecomposable P(6,5,3,1) not known.
# Projective indecomposable P(6,5,4) not known.
# Projective indecomposable P(7,4,3,1) not known.
# Projective indecomposable P(7,5,2,1) not known.
# Projective indecomposable P(7,5,3) not known.
# Projective indecomposable P(7,6,2) not known.
# Projective indecomposable P(8,4,2,1) not known.
# Projective indecomposable P(8,4,3) not known.
# Projective indecomposable P(8,5,2) not known.
# Projective indecomposable P(8,6,1) not known.
# Projective indecomposable P(8,7) not known.
# Projective indecomposable P(9,3,2,1) not known.
# Projective indecomposable P(9,4,2) not known.
# Projective indecomposable P(9,5,1) not known.
# Projective indecomposable P(9,6) not known.
# Projective indecomposable P(10,3,2) not known.
# Projective indecomposable P(10,4,1) not known.
# Projective indecomposable P(10,5) not known.
# Projective indecomposable P(11,3,1) not known.
# Projective indecomposable P(11,4) not known.
# Projective indecomposable P(12,2,1) not known.
# Projective indecomposable P(12,3) not known.
# Projective indecomposable P(13,2) not known.
# Projective indecomposable P(14,1) not known.
# Projective indecomposable P(15) not known.
gap> SizeScreen([80,20]);; MissingIndecomposables(d);
The following projectives are missing from <d>:
    [ 15 ]  [ 14, 1 ]  [ 13, 2 ]  [ 12, 3 ]  [ 12, 2, 1 ]  [ 11, 4 ]
[ 11, 3, 1 ]  [ 10, 5 ]  [ 10, 4, 1 ]  [ 10, 3, 2 ]  [ 9, 6 ]  [ 9, 5, 1 ]
[ 9, 4, 2 ]  [ 9, 3, 2, 1 ]  [ 8, 7 ]  [ 8, 6, 1 ]  [ 8, 5, 2 ]  [ 8, 4, 3 ]
[ 8, 4, 2, 1 ]  [ 7, 6, 2 ]  [ 7, 5, 3 ]  [ 7, 5, 2, 1 ]  [ 7, 4, 3, 1 ]
[ 6, 5, 4 ]  [ 6, 5, 3, 1 ]  [ 6, 4, 3, 2 ]

Actually, you are much better starting with the decomposition matrix of S_14 and then applying InducedDecompositionMatrix (3.5-1) to this matrix. See also DecompositionMatrix (3.2-8). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.5-7 MatrixDecompositionMatrix

Returns: the GAP matrix corresponding to the Hecke decomposition matrix d

The rows and columns of d are sorted by the ordering stored in the internal algebra object of the matrix d.

gap> SizeScreen([80,20]);;
gap> MatrixDecompositionMatrix(DecompositionMatrix(Specht(3),5));
[ [ 1, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 1, 1, 0, 0 ], [ 0, 0, 0, 1, 0 ],
  [ 1, 0, 0, 0, 1 ], [ 0, 0, 0, 0, 1 ], [ 0, 0, 1, 0, 0 ] ]

See also DecompositionMatrix (3.2-8) and DecompositionMatrixMatrix (3.5-8). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.5-8 DecompositionMatrixMatrix

Returns: the Hecke decomposition matrix corresponding to the GAP matrix m

If p is the number of partitions of n and r the number of e-regular partitions of n, then m must be either r× r, p× r or p× p. The rows and columns of m are assumed to be indexed by partitions sorted by the ordering stored in the algebra object H (see Specht (3.2-1)).

gap> H:=Specht(3);;
gap> m:=[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 1, 0 ],
>         [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ] ];;
gap> Display(DecompositionMatrixMatrix(H,m,4));
4    | 1
3,1  | . 1
2^2  | 1 . 1
2,1^2| . . . 1
1^4  | . . 1 .

See also DecompositionMatrix (3.2-8) and MatrixDecompositionMatrix (3.5-7). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.5-9 AddIndecomposable

AddIndecomposable inserts the indecomposable module x into the decomposition matrix d. If d already contains the indecomposable d then a warning is printed. The function AddIndecomposable also calculates MullineuxMap( x) (see MullineuxMap (3.7-3)) and adds this indecomposable to d (or checks to see that it agrees with the corresponding entry of d if this indecomposable is already in d).

See IsNewIndecomposable (3.5-2) for an example. See also DecompositionMatrix (3.2-8) and CrystalDecompositionMatrix (3.2-9). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.5-10 RemoveIndecomposable

The function RemoveIndecomposable removes the column from d which corresponds to P(μ). This is sometimes useful when trying to calculate a new decomposition matrix using Hecke and want to test a possible candidate for a yet to be identified PIM.

See also DecompositionMatrix (3.2-8) and CrystalDecompositionMatrix (3.2-9). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.5-11 MissingIndecomposables

The function MissingIndecomposables prints the list of partitions corresponding to the indecomposable modules which are not listed in d.

See also DecompositionMatrix (3.2-8) and CrystalDecompositionMatrix (3.2-9). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.6 Calculating dimensions

Hecke has two functions for calculating the dimensions of modules of Hecke algebras; SimpleDimension (3.6-1) and SpechtDimension (3.6-2). As yet, Hecke does not know how to calculate the dimensions of modules for q-Schur algebras (these depend up on q).

3.6-1 SimpleDimension

In the first two forms, SimpleDimension prints the dimensions of all of the simple modules specified by d or for the Hecke algebra H(S_n) respectively. If a partition μ is supplied, as in the last form, then the dimension of the simple module D(μ) is returned. At present the function is not implemented for the simple modules of the q-Schur algebras.

gap> H:=Specht(6);;
gap> SimpleDimension(H,11,3);
272
gap> d:=DecompositionMatrix(H,5);; SimpleDimension(d,3,2);
5
gap> SimpleDimension(d);
5     : 1
4,1   : 4
3,2   : 5
3,1^2 : 6
2^2,1 : 5
2,1^3 : 4
1^5   : 1
true

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.6-2 SpechtDimension

Returns: the dimension of the Specht module S(μ)

dim S(μ) is equal to the number of standard μ-tableaux; the answer is given by the hook length formula (see [JK81]).

gap> SpechtDimension(6,3,2,1);
5632

See also SimpleDimension (3.6-1). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7 Combinatorics on Young diagrams

These functions range from the representation theoretic q-Schaper theorem and Kleshchev's algorithm for the Mullineux map through to simple combinatorial operations like adding and removing rim hooks from Young diagrams.

3.7-1 Schaper

Returns: a linear combination of Specht modules which have the same composition factors as the sum of the modules in the "Jantzen filtration " of S(μ); see [JM97]. In particular, if ν strictly dominates μ then D(ν) is a composition factor of S(μ) if and only if it is a composition factor of Schaper(μ).

Schaper uses the valuation map attached to H (see Specht (3.2-1) and [JM97]).

One way in which the q-Schaper theorem can be applied is as follows. Suppose that we have a projective module x, written as a linear combination of Specht modules and suppose that we are trying to decide whether the projective indecomposable P(μ) is a direct summand of x. Then, providing that we know that P(ν) is not a summand of x for all (e-regular) partitions ν which strictly dominate μ (see Dominates (3.8-12)), P(μ) is a summand of x if and only if InnerProduct(Schaper( H,μ),x) is non-zero (note, in particular, that we don't need to know the indecomposable P(μ) in order to perform this calculation).

The q-Schaper theorem can also be used to check for irreduciblity; in fact, this is the basis for the criterion employed by IsSimpleModule (3.7-2).

gap> SizeScreen([80,20]);; H:=Specht(2);;
gap> Display(Schaper(H,9,5,3,2,1));
S(17,2,1) - S(15,2,1^3) + S(13,2^3,1) - S(11,3^2,2,1) + S(10,4,3,2,1) - S(9,8,\
3) - S(9,8,1^3) + S(9,6,3,2) + S(9,6,3,1^2) + S(9,6,2^2,1)
gap> Display(Schaper(H,9,6,5,2));
0S()

The last calculation shows that S(9,6,5,2) is irreducible when R is a field of characteristic zero and e=2 (cf. IsSimpleModule(H,9,6,5,2)). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-2 IsSimpleModule

Returns: true if S(μ) is simple and false otherwise.

μ an e-regular partition.

This calculation uses the valuation function of H; see Specht (3.2-1). Note that the criterion used by IsSimpleModule is completely combinatorial; it is derived from the q-Schaper theorem [JM97].

gap> H:=Specht(3);;
gap> IsSimpleModule(H,45,31,24);
false

See also Schaper (3.7-1). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-3 MullineuxMap

The sign representation D(1^n) of the Hecke algebra is the (one dimensional) representation sending T_w to (-1)^l(w). The Hecke algebra H is not a Hopf algebra so there is no well defined action of H upon the tensor product of two H-modules; however, there is an outer automorphism # of H which corresponds to tensoring with D(1^n). This sends an irreducible module D(μ) to an irreducible D(μ)^#≅ D(μ^#) for some e-regular partition μ^#. In the symmetric group case, Mullineux gave a conjectural algorithm for calculating μ^#; consequently the map sending μ to μ^# is known as the Mullineux map.

Deep results of Kleshchev [Kle96] for the symmetric group give another (proven) algorithm for calculating the partition μ^# (Ford and Kleshchev have deduced Mullineux's conjecture from this). Using the canonical basis, it was shown by [LLT96] that the natural generalization of Kleshchev's algorithm to H gives the Mullineux map for Hecke algebras over fields of characteristic zero. The general case follows from this, so the Mullineux map is now known for all Hecke algebras.

Kleshchev's map is easy to describe; he proved that if gns is any good node sequence for μ, then the sequence obtained from gns by replacing each residue r by -rmod e is a good node sequence for μ^# (see GoodNodeSequence (3.7-8)).

gap> MullineuxMap(Specht(2),12,5,2);
[ 12, 5, 2 ]
gap> MullineuxMap(Specht(4),12,5,2);
[ 4, 4, 4, 2, 2, 1, 1, 1 ]
gap> MullineuxMap(Specht(6),12,5,2);
[ 4, 3, 2, 2, 2, 2, 2, 1, 1 ]
gap> MullineuxMap(Specht(8),12,5,2);
[ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 ]
gap> MullineuxMap(Specht(10),12,5,2);
[ 3, 3, 3, 3, 2, 1, 1, 1, 1, 1 ]

Returns: the image of μ under the Mullineux map

The Mullineux map can also be calculated using a decomposition matrix. To see this recall that "tensoring" a Specht module S(μ) with the sign representation yields a module isomorphic to the dual of S(λ), where λ is the partition conjugate to μ. It follows that d_μ,ν=d_λ,ν^# for all e-regular partitions ν. Therefore, if μ is the last partition in the lexicographic order such that d_μ,ν≠ 0 then we must have ν^#=λ. The second form of MullineuxMap uses d to calculate μ^# rather than the Kleshchev-[LLT96] result.

Returns: returns x^#, the image of x under #.

Note that the above remarks show that P(μ) is mapped to P(μ^#) via the Mullineux map; this observation is useful when calculating decomposition matrices (and is used by the function InducedDecompositionMatrix (3.5-1)).

See also GoodNodes (3.7-6) and GoodNodeSequence (3.7-8). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-4 MullineuxSymbol

Returns: the Mullineux symbol of the e-regular partition μ.

gap> MullineuxSymbol(5,[8,6,5,5]);
[ [ 10, 6, 5, 3 ], [ 4, 4, 3, 2 ] ]

See also PartitionMullineuxSymbol (3.7-5). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-5 PartitionMullineuxSymbol

Returns: the e-regular partition corresponding to the given Mullineux symbol ms

gap> PartitionMullineuxSymbol(5, MullineuxSymbol(5,[8,6,5,5]) );
[ 8, 6, 5, 5 ]

See also MullineuxSymbol (3.7-4). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-6 GoodNodes

Returns: a list of the rows of μ which end in a good node. The good node of residue r (if it exists) is the (r+1)-st element in this list.

Returns: the number of the row which ends with the good node of residue r or fail if there is no good node of residue r.

Given a partition and an integer e, Kleshchev [K] defined the notion of good node for each residue r (0≤ r<e). When e is prime and μ is e-regular, Kleshchev showed that the good nodes describe the restriction of the socle of D(μ) in the symmetric group case. Brundan [Bru98] has recently generalized this result to the Hecke algebra.

By definition, there is at most one good node for each residue r and this node is a removable node (in the diagram of μ).

gap> GoodNodes(5,[5,4,3,2]);
[ fail, fail, 2, fail, 1 ]
gap> GoodNodes(5,[5,4,3,2],0);
fail
gap> GoodNodes(5,[5,4,3,2],4);
1

The good nodes also determine the Kleshchev-Mullineux map (see GoodNodeSequence (3.7-8) and MullineuxMap (3.7-3)). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-7 NormalNodes

Returns: the numbers of the rows of μ which end in one of Kleshchev's [Kle96] normal nodes.

Returns: the rows corresponding to normal nodes of the specified residue.

gap> NormalNodes(5,[6,5,4,4,3,2,1,1,1]);
[ [ 1, 4 ], [  ], [  ], [ 2, 5 ], [  ] ]
gap> NormalNodes(5,[6,5,4,4,3,2,1,1,1],0);
[ 1, 4 ]

See also GoodNodes (3.7-6). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-8 GoodNodeSequence

Given an e-regular partition μ of n, a good node sequence for μ is a sequence gns of n residues such that μ has a good node of residue r, where r is the last residue in gns and the first n-1 residues in gns are a good node sequence for the partition obtained from μ by deleting its (unique) good node with residue r (see GoodNodes (3.7-6)). In general, μ will have more than one good node sequence; however, any good node sequence uniquely determines μ (see PartitionGoodNodeSequence (3.7-9)).

gap> H:=Specht(4);; GoodNodeSequence(H,4,3,1);
[ 0, 3, 1, 0, 2, 2, 1, 3 ]
gap> GoodNodeSequence(H,4,3,2);
[ 0, 3, 1, 0, 2, 2, 1, 3, 3 ]
gap> GoodNodeSequence(H,4,4,2);
[ 0, 3, 1, 0, 2, 2, 1, 3, 3, 2 ]
gap> GoodNodeSequence(H,5,4,2);
[ 0, 3, 1, 0, 2, 2, 1, 3, 3, 2, 0 ]

Returns: list of all good node sequences for μ

gap> H:=Specht(4);; GoodNodeSequences(H,5,2,1);
[ [ 0, 1, 2, 3, 3, 2, 0, 0 ], [ 0, 3, 1, 2, 2, 3, 0, 0 ],
  [ 0, 1, 3, 2, 2, 3, 0, 0 ], [ 0, 1, 2, 3, 3, 0, 2, 0 ],
  [ 0, 1, 2, 3, 0, 3, 2, 0 ], [ 0, 1, 2, 3, 3, 0, 0, 2 ],
  [ 0, 1, 2, 3, 0, 3, 0, 2 ] ]

The good node sequences determine the Mullineux map (see GoodNodes (3.7-6) and MullineuxMap (3.7-3)). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-9 PartitionGoodNodeSequence

Returns: the unique e-regular partition corresponding to gns (or fail if in fact gns is not a good node sequence).

gap> H:=Specht(4);;
gap> PartitionGoodNodeSequence(H,0, 3, 1, 0, 2, 2, 1, 3, 3, 2);
[ 4, 4, 2 ]

See also GoodNodes (3.7-6), GoodNodeSequence (3.7-8) and MullineuxMap (3.7-3). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-10 GoodNodeLatticePath

Returns: a sequence of partitions which give a path in the e-good partition lattice from the empty partition to μ.

Returns: the list of all paths in the e-good partition lattice which end in μ.

Returns: the path corresponding to a given good node sequence gns

gap> GoodNodeLatticePath(3,3,2,1);
[ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 1, 1 ], [ 2, 2, 1 ], [ 3, 2, 1 ] ]
gap> GoodNodeLatticePaths(3,3,2,1);
[ [ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 1, 1 ], [ 2, 2, 1 ], [ 3, 2, 1 ] ],
  [ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 2 ], [ 2, 2, 1 ], [ 3, 2, 1 ] ] ]
gap> GoodNodeSequence(4,6,3,2);
[ 0, 3, 1, 0, 2, 2, 3, 3, 0, 1, 1 ]
gap> LatticePathGoodNodeSequence(4,last);
[ [ 1 ], [ 1, 1 ], [ 2, 1 ], [ 2, 2 ], [ 3, 2 ], [ 3, 2, 1 ], [ 4, 2, 1 ],
  [ 4, 2, 2 ], [ 5, 2, 2 ], [ 6, 2, 2 ], [ 6, 3, 2 ] ]

See also GoodNodes (3.7-6). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-11 LittlewoodRichardsonRule

Given partitions μ of n and ν of m the module S(μ)⊗ S(ν) is naturally an H(S_n× S_m)- module and, by inducing, we obtain an H(S_n+m)-module. This module has the same composition factors as ∑_ν a_μ,ν^λ S(λ), where the sum runs over all partitions λ of n+m and the integers a_μ,ν^λ are the Littlewood-Richardson coefficients. The integers a_μ,ν^λ can be calculated using a straightforward combinatorial algorithm known as the Littlewood-Richardson rule (see [JK81]). The function LittlewoodRichardsonRule returns an (unordered) list of partitions of n+m in which each partition λ occurs a_μ,ν^λ times. The Littlewood-Richardson coefficients are independent of e; they can be read more easily from the computation S(μ)⊗ S(ν).

gap> SizeScreen([80,20]);;
gap> H:=Specht(0);; # the generic Hecke algebra with R=C[q]
gap> LittlewoodRichardsonRule([3,2,1],[4,2]);
[ [ 4, 3, 2, 2, 1 ], [ 4, 3, 3, 1, 1 ], [ 4, 3, 3, 2 ], [ 4, 4, 2, 1, 1 ],
  [ 4, 4, 2, 2 ], [ 4, 4, 3, 1 ], [ 5, 2, 2, 2, 1 ], [ 5, 3, 2, 1, 1 ],
  [ 5, 3, 2, 2 ], [ 5, 4, 2, 1 ], [ 5, 3, 2, 1, 1 ], [ 5, 3, 3, 1 ],
  [ 5, 4, 1, 1, 1 ], [ 5, 4, 2, 1 ], [ 5, 5, 1, 1 ], [ 5, 3, 2, 2 ],
  [ 5, 3, 3, 1 ], [ 5, 4, 2, 1 ], [ 5, 4, 3 ], [ 5, 5, 2 ], [ 6, 2, 2, 1, 1 ],
  [ 6, 3, 1, 1, 1 ], [ 6, 3, 2, 1 ], [ 6, 4, 1, 1 ], [ 6, 2, 2, 2 ],
  [ 6, 3, 2, 1 ], [ 6, 4, 2 ], [ 6, 3, 2, 1 ], [ 6, 3, 3 ], [ 6, 4, 1, 1 ],
  [ 6, 4, 2 ], [ 6, 5, 1 ], [ 7, 2, 2, 1 ], [ 7, 3, 1, 1 ], [ 7, 3, 2 ],
  [ 7, 4, 1 ] ]
gap> Display(MakeSpecht(H,3,2,1)*MakeSpecht(H,4,2));
S(7,4,1) + S(7,3,2) + S(7,3,1^2) + S(7,2^2,1) + S(6,5,1) + 2S(6,4,2) + 2S(6,4,\
1^2) + S(6,3^2) + 3S(6,3,2,1) + S(6,3,1^3) + S(6,2^3) + S(6,2^2,1^2) + S(5^2,2\
) + S(5^2,1^2) + S(5,4,3) + 3S(5,4,2,1) + S(5,4,1^3) + 2S(5,3^2,1) + 2S(5,3,2^\
2) + 2S(5,3,2,1^2) + S(5,2^3,1) + S(4^2,3,1) + S(4^2,2^2) + S(4^2,2,1^2) + S(4\
,3^2,2) + S(4,3^2,1^2) + S(4,3,2^2,1)
gap> LittlewoodRichardsonCoefficient([3,2,1],[4,2],[5,4,2,1]);
3

The function LittlewoodRichardsonCoefficient returns a single Littlewood-Richardson coefficient (although you are really better off asking for all of them, since they will all be calculated anyway).

See also RInducedModule (3.4-1) and InverseLittlewoodRichardsonRule (3.7-12). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-12 InverseLittlewoodRichardsonRule

Returns: a list of all pairs of partitions [μ,ν] such that the Littlewood-Richardson coefficient a_μ,ν^τ is non-zero (see LittlewoodRichardsonRule (3.7-11)). The list returned is unordered and [μ,ν] will appear a_μ,ν^τ times in it.

gap> SizeScreen([80,20]);; InverseLittlewoodRichardsonRule(3,2,1);
[ [ [  ], [ 3, 2, 1 ] ], [ [ 1 ], [ 3, 2 ] ], [ [ 1 ], [ 2, 2, 1 ] ],
  [ [ 1 ], [ 3, 1, 1 ] ], [ [ 1, 1 ], [ 2, 2 ] ], [ [ 1, 1 ], [ 3, 1 ] ],
  [ [ 1, 1 ], [ 2, 1, 1 ] ], [ [ 1, 1, 1 ], [ 2, 1 ] ], [ [ 2 ], [ 2, 2 ] ],
  [ [ 2 ], [ 3, 1 ] ], [ [ 2 ], [ 2, 1, 1 ] ], [ [ 2, 1 ], [ 3 ] ],
  [ [ 2, 1 ], [ 2, 1 ] ], [ [ 2, 1 ], [ 2, 1 ] ], [ [ 2, 1 ], [ 1, 1, 1 ] ],
  [ [ 2, 1, 1 ], [ 2 ] ], [ [ 2, 1, 1 ], [ 1, 1 ] ], [ [ 2, 2 ], [ 2 ] ],
  [ [ 2, 2 ], [ 1, 1 ] ], [ [ 2, 2, 1 ], [ 1 ] ], [ [ 3 ], [ 2, 1 ] ],
  [ [ 3, 1 ], [ 2 ] ], [ [ 3, 1 ], [ 1, 1 ] ], [ [ 3, 1, 1 ], [ 1 ] ],
  [ [ 3, 2 ], [ 1 ] ], [ [ 3, 2, 1 ], [  ] ] ]

See also LittlewoodRichardsonRule (3.7-11). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-13 EResidueDiagram

The e-residue of the (i,j)-th node in the diagram of a partition μ is (j-i)mod e. EResidueDiagram(e, μ) prints the diagram of the partition μ replacing each node with its e-residue. If x is a module then EResidueDiagram(x) prints the e-residue diagrams of all of the e-regular partitions appearing in x (such diagrams are useful when trying to decide how to restrict and induce modules and also in applying results such as the "Scattering theorem" of [JM96]). It is not necessary to supply the integer e in this case because x "knows" the value of e.

gap> H:=Specht(2);; EResidueDiagram(MakeSpecht(MakePIM(H,7,5)));
[ 7, 5 ]
   0   1   0   1   0   1   0
   1   0   1   0   1
[ 6, 5, 1 ]
   0   1   0   1   0   1
   1   0   1   0   1
   0
[ 5, 4, 2, 1 ]
   0   1   0   1   0
   1   0   1   0
   0   1
   1
# There are 3 2-regular partitions.
true

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-14 HookLengthDiagram

Prints the diagram of μ, replacing each node with its hook length (see [JK81]).

gap> HookLengthDiagram(11,6,3,2);
  14  13  11   9   8   7   5   4   3   2   1
   8   7   5   3   2   1
   4   3   1
   2   1
true

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-15 RemoveRimHook

Returns: the partition obtained from μ by removing the (row, col)-th rim hook from (the diagram of) μ.

gap> RemoveRimHook([6,5,4],1,2);
[ 4, 3, 1 ]
gap> RemoveRimHook([6,5,4],2,3);
[ 6, 3, 2 ]
gap> HookLengthDiagram(6,5,4);
   8   7   6   5   3   1
   6   5   4   3   1
   4   3   2   1
true

See also AddRimHook (3.7-16). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.7-16 AddRimHook

Returns: a list [ν, l] where ν is the partition obtained from μ by adding a rim hook of length h with its "foot" in the r-th row of (the diagram of) μ and l is the leg length of the wrapped on rim hook (see, for example, [JK81]). If the resulting diagram ν is not the diagram of a partition then fail is returned.

gap> AddRimHook([6,4,3],1,3);
[ [ 9, 4, 3 ], 0 ]
gap> AddRimHook([6,4,3],2,3);
fail
gap> AddRimHook([6,4,3],3,3);
[ [ 6, 5, 5 ], 1 ]
gap> AddRimHook([6,4,3],4,3);
[ [ 6, 4, 3, 3 ], 0 ]
gap> AddRimHook([6,4,3],5,3);
fail

See also RemoveRimHook (3.7-15). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8 Operations on partitions

This section contains functions for manipulating partitions and also several useful orderings on the set of partitions.

3.8-1 ECore

Returns: the e-core of the partition μ.

The e-core of a partition μ is what remains after as many rim e-hooks as possible have been removed from the diagram of μ (that this is well defined is not obvious; see [JK81]).

gap> H:=Specht(6);; ECore(H,16,8,6,5,3,1);
[ 4, 3, 1, 1 ]

The e-core is calculated here using James'; notation of an abacus there is also an EAbacus function; but it is more "pretty" than useful.

See also IsECore (3.8-2), EQuotient (3.8-3) and EWeight (3.8-5). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-2 IsECore

Returns: true if μ is an e-core and false otherwise.

See also ECore (3.8-1). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-3 EQuotient

Returns: the e-quotient of μ; this is a sequence of e partitions whose definition can be found in [JK81].

gap> H:=Specht(8);; EQuotient(H,22,18,16,12,12,1,1);
[ [ 1, 1 ], [  ], [  ], [  ], [  ], [ 2, 2 ], [  ], [ 1 ] ]

See also ECore (3.8-1) and CombineEQuotientECore (3.8-4). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-4 CombineEQuotientECore

Returns: the partition which has e-quotient q and e -core C.

A partition is uniquely determined by its e-quotient and its e -core (see EQuotient (3.8-3) and ECore (3.8-1)).

gap> H:=Specht(11);; mu:=[100,98,57,43,12,1];;
gap> Q:=EQuotient(H,mu);
[ [ 9 ], [  ], [  ], [  ], [  ], [  ], [ 3 ], [ 1 ], [ 9 ], [  ], [ 5 ] ]
gap> C:=ECore(H,mu);
[ 7, 2, 2, 1, 1, 1 ]
gap> CombineEQuotientECore(H,Q,C);
[ 100, 98, 57, 43, 12, 1 ]

See also ECore (3.8-1) and EQuotient (3.8-3). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-5 EWeight

The e-weight of a partition is the number of e-hooks which must be removed from the partition to reach the e-core (see ECore (3.8-1)).

gap> EWeight(6,[16,8,6,5,3,1]);
5

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-6 ERegularPartitions

Returns: the list of e-regular partitions of n, ordered reverse lexicographically (see Lexicographic (3.8-14)).

A partition μ=(μ_1,μ_2,...) is e-regular if there is no integer i such that μ_i=μ_i+1=⋯=μ_i+e-1>0.

gap> H:=Specht(3);; ERegularPartitions(H,6);
[ [ 2, 2, 1, 1 ], [ 3, 2, 1 ], [ 3, 3 ], [ 4, 1, 1 ], [ 4, 2 ], [ 5, 1 ],
  [ 6 ] ]

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-7 IsERegular

Returns: true if μ is e-regular and false otherwise.

This functions requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-8 ConjugatePartition

Returns: the partition whose diagram is obtained by interchanging the rows and columns in the diagram of μ.

gap> ConjugatePartition(6,4,3,2);
[ 4, 4, 3, 2, 1, 1 ]

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-9 BetaSet

Returns: a set of beta numbers (i.e. first column hook lengths; see [JK81]) corresponding to the partition μ.

gap> BetaSet([5,4,2,2]);
[ 2, 3, 6, 8 ]

See also PartitionBetaSet (3.8-10). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-10 PartitionBetaSet

Returns: the partitions corresponding to the given set of beta numbers bn.Note in particular that bn must be a set of integers.

gap> PartitionBetaSet([ 2, 3, 6, 8 ]);
[ 5, 4, 2, 2 ]

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-11 ETopLadder

The ladders in the diagram of a partition are the lines connecting nodes of constant e-residue, having slope e-1 (see [JK81]). A new partition can be obtained from μ by sliding all nodes up to the highest possible rungs on their ladders.

Returns: the partition obtained in this way; it is automatically e-regular (this partition is denoted μ^R in [JK81]).

gap> H:=Specht(4);;
gap> ETopLadder(H,1,1,1,1,1,1,1,1,1,1);
[ 4, 3, 3 ]
gap> ETopLadder(6,1,1,1,1,1,1,1,1,1,1);
[ 2, 2, 2, 2, 2 ]

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-12 Dominates

Returns: true if either μ=ν or ∀ i≥ 1: ∑_j=1^iμ_j≥∑_j=1^iν_j and false otherwise.

The dominance ordering is an important partial order in the representation theory of Hecke algebra because d_μ,ν=0 unless ν dominates μ.

gap> Dominates([5,4],[4,4,1]);
true

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-13 LengthLexicographic

Returns: true if the length of μ is less than the length of ν or if the length of μ equals the length of ν and Lexicographic(μ, ν).

gap> p:=Partitions(6);;Sort(p,LengthLexicographic); p;
[ [ 6 ], [ 5, 1 ], [ 4, 2 ], [ 3, 3 ], [ 4, 1, 1 ], [ 3, 2, 1 ], [ 2, 2, 2 ],
  [ 3, 1, 1, 1 ], [ 2, 2, 1, 1 ], [ 2, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 1 ] ]

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-14 Lexicographic

Returns: true if μ is lexicographically greater than or equal to ν.

gap> p:=Partitions(6);;Sort(p,Lexicographic); p;
[ [ 6 ], [ 5, 1 ], [ 4, 2 ], [ 4, 1, 1 ], [ 3, 3 ], [ 3, 2, 1 ],
  [ 3, 1, 1, 1 ], [ 2, 2, 2 ], [ 2, 2, 1, 1 ], [ 2, 1, 1, 1, 1 ],
  [ 1, 1, 1, 1, 1, 1 ] ]

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.8-15 ReverseDominance

Returns: true if ∀ i>0: ∑_j≥ iμ_j > ∑_j≥ iν_j.

This is another total order on partitions which extends the dominance ordering (see Dominates (3.8-12)).

gap> p:=Partitions(6);;Sort(p,ReverseDominance); p;
[ [ 6 ], [ 5, 1 ], [ 4, 2 ], [ 3, 3 ], [ 4, 1, 1 ], [ 3, 2, 1 ], [ 2, 2, 2 ],
  [ 3, 1, 1, 1 ], [ 2, 2, 1, 1 ], [ 2, 1, 1, 1, 1 ], [ 1, 1, 1, 1, 1, 1 ] ]

This is the ordering used by James in the appendix of his Springer lecture notes book.

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.9 Miscellaneous functions on modules

This section contains some functions for looking at the partitions in a given module for the Hecke algebras. Most of them are used internally by Hecke.

3.9-1 Specialized

Returns: the corresponding element of the Grothendieck ring or the corresponding decomposition matrix of the Hecke algebra when given an element of the Fock space x (see Specht (3.2-1)), or a crystallized decomposition matrix (see CrystalDecompositionMatrix (3.2-9)), respectively.

By default the indeterminate v is specialized to 1; however v can be specialized to any (integer) q by supplying a second argument.

gap> SizeScreen([80,20]);; H:=Specht(2);; x:=MakeFockPIM(H,6,2);; Display(x);
Sq(6,2) + vSq(6,1^2) + vSq(5,3) + v^2Sq(5,1^3) + vSq(4,3,1) + v^2Sq(4,2^2) + (\
v^3+v)Sq(4,2,1^2) + v^2Sq(4,1^4) + v^2Sq(3^2,1^2) + v^3Sq(3,2^2,1) + v^3Sq(3,1\
^5) + v^3Sq(2^3,1^2) + v^4Sq(2^2,1^4)
gap> Display(Specialized(x));
S(6,2) + S(6,1^2) + S(5,3) + S(5,1^3) + S(4,3,1) + S(4,2^2) + 2S(4,2,1^2) + S(\
4,1^4) + S(3^2,1^2) + S(3,2^2,1) + S(3,1^5) + S(2^3,1^2) + S(2^2,1^4)
gap> Display(Specialized(x,2));
S(6,2) + 2S(6,1^2) + 2S(5,3) + 4S(5,1^3) + 2S(4,3,1) + 4S(4,2^2) + 10S(4,2,1^2\
) + 4S(4,1^4) + 4S(3^2,1^2) + 8S(3,2^2,1) + 8S(3,1^5) + 8S(2^3,1^2) + 16S(2^2,\
1^4)

An example of Specialized being applied to a crystallized decomposition matrix can be found in CrystalDecompositionMatrix (3.2-9). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.9-2 ERegulars

ERegulars(x) prints a list of the e-regular partitions, together with multiplicities, which occur in the module x. ListERegulars(x) returns an actual list of these partitions rather than printing them.

gap> H:=Specht(8);;
gap> x:=MakeSpecht(RInducedModule(MakePIM(H,8,5,3)));; Display(x);
S(9,5,3) + S(8,6,3) + S(8,5,4) + S(8,5,3,1) + S(6,5,3^2) + S(5^2,4,3) + S(5^2,3^2,1)
gap> ERegulars(x);
[ 9, 5, 3 ]  [ 8, 6, 3 ]  [ 8, 5, 4 ]  [ 8, 5, 3, 1 ]
[ 6, 5, 3, 3 ]  [ 5, 5, 4, 3 ]  [ 5, 5, 3, 3, 1 ]
gap> Display(MakePIM(x));
P(9,5,3) + P(8,6,3) + P(8,5,4) + P(8,5,3,1)

This example shows why these functions are useful: given a projective module x, as aboveand the list of e-regular partitions in x we know the possible indecomposable direct summands of x.

Note that it is not necessary to specify what e is when calling this function because x "knows" the value of e.

The function ERegulars can also be applied to a decomposition matrix d; in this case it returns the unitriangular submatrix of d whose rows and columns are indexed by the e-regular partitions.

These function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.9-3 SplitECores

Returns: a list [b_1,...,b_k] where the Specht modules in each b_i all belong to the same block (i.e. they have the same e-core).

Returns: the component of x which is in the same block as μ.

Returns: the component of x which is in the same block as y.

gap> H:=Specht(2);;
gap> Display(SplitECores(RInducedModule(MakeSpecht(H,5,3,1))));
[ S(6,3,1) + S(5,3,2) + S(5,3,1,1), S(5,4,1) ]
gap> Display(RInducedModule(MakeSpecht(H,5,3,1),0));
S(5,4,1)
gap> Display(RInducedModule(MakeSpecht(H,5,3,1),1));
S(6,3,1) + S(5,3,2) + S(5,3,1^2)

See also ECore (3.8-1), RInducedModule (3.4-1) and RRestrictedModule (3.4-3). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.9-4 Coefficient

Returns: the coefficient of S(μ) in x (resp. D(μ), or P(μ)).

gap> SizeScreen([80,20]);;
gap> H:=Specht(3);; x:=MakeSpecht(MakePIM(H,7,3));; Display(x);
S(7,3) + S(7,2,1) + S(6,2,1^2) + S(5^2) + S(5,2^2,1) + S(4^2,1^2) + S(4,3^2) +\
 S(4,3,2,1)
gap> Coefficient(x,5,2,2,1);
1

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.9-5 InnerProduct

Here x and y are some modules of the Hecke algebra (i.e. Specht modules, PIMS, or simple modules). InnerProduct computes the standard inner product of these elements. This is sometimes a convenient way to compute decomposition numbers (for example).

gap> H:=Specht(2);; InnerProduct(MakeSpecht(H,2,2,2,1), MakePIM(H,4,3));
1
gap> DecompositionNumber(H,[2,2,2,1],[4,3]);
1

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.10 Semi-standard and standard tableaux

These functions are not really part of Hecke proper; however they are related and may well be of use to someone. Tableaux are represented by objects, that can be constructed from a list of lists.

3.10-1 Tableau

Returns: tableau object corresponding to the given list of lists

This is the constructor for tableau objects. The first entry of the given argument list is the list corresponding to the first row of the tableau.

3.10-2 SemiStandardTableaux

Returns: list of the semistandard μ-tableaux of type ν [JK81]

μ a partition, ν a composition.

gap> SizeScreen([80,20]);; Display(SemiStandardTableaux([4,3],[1,1,1,2,2]));
[ Tableau( [ [ 1, 2, 3, 4 ], [ 4, 5, 5 ] ] ),
  Tableau( [ [ 1, 2, 3, 5 ], [ 4, 4, 5 ] ] ),
  Tableau( [ [ 1, 2, 4, 4 ], [ 3, 5, 5 ] ] ),
  Tableau( [ [ 1, 2, 4, 5 ], [ 3, 4, 5 ] ] ),
  Tableau( [ [ 1, 3, 4, 4 ], [ 2, 5, 5 ] ] ),
  Tableau( [ [ 1, 3, 4, 5 ], [ 2, 4, 5 ] ] ) ]

See also StandardTableaux (3.10-3). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.10-3 StandardTableaux

Returns: list of the standard μ-tableaux

μ a partition

gap> SizeScreen([80,20]);; Display(StandardTableaux(4,2));
[ Tableau( [ [ 1, 2, 3, 4 ], [ 5, 6 ] ] ), Tableau( [ [ 1, 2, 3, 5 ], [ 4, 6 ] ] ),
  Tableau( [ [ 1, 2, 3, 6 ], [ 4, 5 ] ] ), Tableau( [ [ 1, 2, 4, 5 ], [ 3, 6 ] ] ),
  Tableau( [ [ 1, 2, 4, 6 ], [ 3, 5 ] ] ), Tableau( [ [ 1, 2, 5, 6 ], [ 3, 4 ] ] ),
  Tableau( [ [ 1, 3, 4, 5 ], [ 2, 6 ] ] ), Tableau( [ [ 1, 3, 4, 6 ], [ 2, 5 ] ] ),
  Tableau( [ [ 1, 3, 5, 6 ], [ 2, 4 ] ] ) ]

See also SemiStandardTableaux (3.10-2). This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.10-4 ConjugateTableau

Returns: tableau obtained from tab by interchangings its rows and columns

gap> Display(ConjugateTableau(Tableau([ [ 1, 3, 5, 6 ], [ 2, 4 ] ])));
Standard Tableau:
1	2
3	4
5
6

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.10-5 ShapeTableau

Returns: the partition (or composition) obtained from tab

gap> ShapeTableau( Tableau([ [ 1, 1, 2, 3 ], [ 4, 5 ] ]) );
[ 4, 2 ]

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

3.10-6 TypeTableau

Returns: the type of the (semistandard) tableau tab

The type of a tableau is, the composition σ= (σ_1,σ_2,...) where σ_i is the number of entries in tab which are equal to i.

gap> SizeScreen([80,20]);;
gap> List(SemiStandardTableaux([5,4,2],[4,3,0,1,3]),TypeTableau);
[ [ 4, 3, 0, 1, 3 ], [ 4, 3, 0, 1, 3 ], [ 4, 3, 0, 1, 3 ], [ 4, 3, 0, 1, 3 ],
  [ 4, 3, 0, 1, 3 ] ]

This function requires the package hecke (see LoadPackage (Reference: LoadPackage)).

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