dim(List,SchurRingElement) -- dimension of representation

Function: dim
Usage:

d = dim(lis,s)

d = dim(n,s)

d = dim s
Inputs:
- lis, a list, or Thing
- s, an instance of the type SchurRingElement,
Outputs:
- d, an integer, or Expression

Description

The method returns the dimension of the virtual representation whose character is represented by s.

i1 : S = schurRing(s,3)

o1 = S

o1 : SchurRing

i2 : dim s_2

o2 = 6

i3 : T = schurRing(t,4,GroupActing => "Sn")

o3 = T

o3 : SchurRing

i4 : dim t_{2,2}

o4 = 2

i5 : U = schurRing(T,u,3)

o5 = U

o5 : SchurRing

i6 : dim (t_{2,2}*u_2)

o6 = 12

Schur characters see the rank of the ambient vector space. The representation s_{2,1} of GL(3) is 8-dimensional, while the same shape in GL(4) gives a 20-dimensional representation:

i7 : dim ((schurRing(s3,3))_{2,1})

o7 = 8

i8 : dim ((schurRing(s4,4))_{2,1})

o8 = 20

Dimensions are also computed for symplectic and orthogonal characters. For orthogonal groups the option OddOrEven selects O(2m+1) versus O(2m):

i9 : Sp = schurRing(QQ, sp, 3, GroupActing => "Sp");

i10 : dim sp_{2,1}

o10 = 64

i11 : Oodd = schurRing(QQ, od, 3, GroupActing => "O", OddOrEven => "Odd");

i12 : dim od_{2,1}

o12 = 105

i13 : Oeven = schurRing(QQ, oe, 3, GroupActing => "O", OddOrEven => "Even");

i14 : dim oe_{2,1}

o14 = 64

If S is a SchurRing of level 1, the ring of polynomial representations of some GL(V), it may sometimes be convenient to compute dimensions of GL(V)-representations symbolically, without specifying the dimension of V. Letting n denote the parameter corresponding to dim(V) we have for example:

i15 : S = schurRing(s,3)

o15 = S

o15 : SchurRing

i16 : dim(n,s_2)

      n(n + 1)
o16 = --------
          2

o16 : Expression of class Divide

i17 : dim(n,s_{1,1})

      (n - 1)n
o17 = --------
          2

o17 : Expression of class Divide

i18 : dim(n,s_{2,1})

      (n - 1)n(n + 1)
o18 = ---------------
             3

o18 : Expression of class Divide

Similar calculations make sense over products of general linear groups. The dimensions of the representations can be computed symbolically as functions of a number of parameters equal to the schurLevel of the ring. The parameters corresponding to levels where the group acting is a symmetric group don't have a good interpretation, so they are disregarded in the dimension calculation. The order of the input parameters is the descending order of the schurLevels: in the example below a corresponds to Q, b corresponds to T and c corresponds to S.

i19 : S = schurRing(s,3)

o19 = S

o19 : SchurRing

i20 : T = schurRing(S,t,4)

o20 = T

o20 : SchurRing

i21 : Q = schurRing(T,q,5,GroupActing => "Sn")

o21 = Q

o21 : SchurRing

i22 : dExpr = dim({a,b,c},s_{2}*t_{1,1}*q_{4,1})

      c(c + 1) (b - 1)b
o22 = --------*--------*4
          2        2

o22 : Expression of class Product

i23 : P = QQ[a,b,c]

o23 = P

o23 : PolynomialRing

i24 : value dExpr

       2 2    2       2
o24 = b c  + b c - b*c  - b*c

o24 : P

i25 : dim({1,2,3},s_{2}*t_{1,1}*q_{4,1})

o25 = 24

o25 : QQ

Over a two-level GL\times GL tower the formula factors as a product of dimensions, one per level, in descending schurLevel order:

i26 : A = schurRing(aR,3);

i27 : B = schurRing(A,bR,2);

i28 : dim(aR_{2,1} * bR_{1,1})

o28 = 8

i29 : dim({4,5}, aR_{2,1} * bR_{1,1})

o29 = 240

o29 : QQ

Ways to use this method:

dim(List,SchurRingElement) -- dimension of representation
dim(SchurRingElement)
dim(Thing,SchurRingElement)

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SchurRings.m2:6960:0.