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# tensor(Character,Character) -- tensor product of characters

## Synopsis

• Function: tensor
• Usage:
tensor(c1,c2)
• Inputs:
• Outputs:

## Description

Returns the tensor product of the input characters. The operator ** may be used for the same purpose.

We construct the character of the coinvariant algebra of the symmetric group on 3 variables.

 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : I = ideal(x+y+z,x*y+x*z+y*z,x*y*z) o2 = ideal (x + y + z, x*y + x*z + y*z, x*y*z) o2 : Ideal of R i3 : S3 = symmetricGroupActors R o3 = {| y z x |, | y x z |, | x y z |} o3 : List i4 : A = action(R/I,S3) o4 = QuotientRing with 3 actors o4 : ActionOnGradedModule i5 : a = character(A,0,3) o5 = Character over R (0, {0}) => | 1 1 1 | (0, {1}) => | -1 0 2 | (0, {2}) => | -1 0 2 | (0, {3}) => | 1 -1 1 | o5 : Character

The Gorenstein duality of this character may be observed by tensoring with the character of the sign representation concentrated in degree 3.

 i6 : sign = character(R,3, hashTable { (0,{3}) => matrix{{1,-1,1}} }) o6 = Character over R (0, {3}) => | 1 -1 1 | o6 : Character i7 : dual(a,{1,2,3}) ** sign === a o7 = true