next | previous | forward | backward | up | index | toc

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

## 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

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

 i7 : S = schurRing(s,3) o7 = S o7 : SchurRing i8 : dim(n,s_2) n(n + 1) o8 = -------- 2 o8 : Expression of class Divide i9 : dim(n,s_{1,1}) (n - 1)n o9 = -------- 2 o9 : Expression of class Divide i10 : dim(n,s_{2,1}) (n - 1)n(n + 1) o10 = --------------- 3 o10 : 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.

 i11 : S = schurRing(s,3) o11 = S o11 : SchurRing i12 : T = schurRing(S,t,4) o12 = T o12 : SchurRing i13 : Q = schurRing(T,q,5,GroupActing => "Sn") o13 = Q o13 : SchurRing i14 : dExpr = dim({a,b,c},s_{2}*t_{1,1}*q_{4,1}) c(c + 1) (b - 1)b o14 = --------*--------*4 2 2 o14 : Expression of class Product i15 : P = QQ[a,b,c] o15 = P o15 : PolynomialRing i16 : value dExpr 2 2 2 2 o16 = b c + b c - b*c - b*c o16 : P i17 : dim({1,2,3},s_{2}*t_{1,1}*q_{4,1}) o17 = 24 o17 : QQ