Schur rings

Schur rings

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 Schur function:

i1 : R = Schur 4;

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

i2 : V = R_{1}



o2 = {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 = R_{3}



o4 = {3}



o4 : R

i5 : dim W



o5 = 20

and the third exterior power of V can be obtained using

i6 : U = R_{1,1,1}



o6 = {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 = {2} + {1, 1}



o8 : R

i9 : V^3



o9 = {3} + 2*{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...