A basis for the diagonally signed-symmetric polynomials

2013/03/14 by Gómez, José Manuel
#05E10 #13A50 #13F20 #20C30 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1303.3491

Abstract

Let n>0 be an integer and let B_n denote the hyperoctahedral group of rank n. The group B_n acts on the polynomial ring Q[x₁,...,x_n,y₁,...,y_n] by signed permutations simultaneously on both of the sets of variables x₁,...,x_n and y₁,...,y_n. The invariant ring M^B_n:=Q[x₁,...,x_n,y₁,...,y_n]^B_n is the ring of diagonally signed-symmetric polynomials. In this article we provide an explicit free basis of M^B_n as a module over the ring of symmetric polynomials on both of the sets of variables x₁²,..., x²_n and y₁²,..., y²_n using signed descent monomials.

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