SchurRings -- rings representing irreducible representations of GL(n)

This package make computations in the representation ring of GL(n) possible.

Given a positive integer n, we may define a polynomial ring over ZZ in n variables, whose monomials correspond to the irreducible representations of GL(n), and where multiplication is given by the decomposition of the tensor product of representations

We create such a ring in Macaulay 2 using the schurRing function:

i1 : R = schurRing(s,4);

A monomial represents the irreducible representation with a given highest weight. The standard 4 dimensional representation is

i2 : V = s_{1} o2 = s 1 o2 : R

We may see the dimension of the corresponding irreducible representation using dim:

i3 : dim V o3 = 4

The third symmetric power of V is obtained by

i4 : W = s_{3} o4 = s 3 o4 : R

i5 : dim W o5 = 20

and the third exterior power of V can be obtained using

i6 : U = s_{1,1,1} o6 = s 1,1,1 o6 : R

i7 : dim U o7 = 4

Multiplication of elements corresponds to tensor product of representations. The value is computed using a variant of the Littlewood-Richardson rule.

i8 : V * V o8 = s + s 2 1,1 o8 : R

i9 : V^3 o9 = s + 2s + s 3 2,1 1,1,1 o9 : R

One cannot make quotients of this ring, and Groebner bases and related computations do not work, but I'm not sure what they would mean...