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

Function: tensor
Usage:

tensor(c1,c2)
Inputs:
- c1, an instance of the type Character,
- c2, an instance of the type Character,
Outputs:
- an instance of the type Character,

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 QQ
      
     (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 : signrep = character(R, hashTable { (0,{3}) => matrix{{1,-1,1}} })

o6 = Character over QQ
      
     (0, {3})  |  1  -1  1

o6 : Character

i7 : dual(a,{1,2,3}) ** signrep === a

o7 = true

Synopsis

Usage:

c ^** m
Inputs:
- A, an instance of the type Character,
- m, an integer,
Outputs:
- an instance of the type Character, the m-th tensor power of c

Starting with version 2.5, this package allows nonnegative tensor powers of characters using Character ^** ZZ.

i8 : a ^** 3

o8 = Character over QQ
      
     (0, {0})  |   1   1   1
     (0, {1})  |  -3   0   6
     (0, {2})  |   0   0  18
     (0, {3})  |   8  -3  35
     (0, {4})  |  -6   0  48
     (0, {5})  |  -6   0  48
     (0, {6})  |   8   3  35
     (0, {7})  |   0   0  18
     (0, {8})  |  -3   0   6
     (0, {9})  |   1  -1   1

o8 : Character

Ways to use this method:

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

The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/BettiCharacters.m2:3859:0.