A combinatorial model for the decomposition of multivariate polynomial rings as $S_n$-modules

Abstract

We consider the symmetric group Sn module of the polynomial ring with m sets of n commuting variables and m ′ sets of n anti-commuting variables and show that the multiplicity of an irreducible indexed by the partition λ (a partition of n) is the number of multiset tableaux of shape λ satisfying certain column and row strict conditions. We also present a finite generating set for the ring of Sn invariant polynomials of this ring.

The Bosonic-Fermionic Diagonal Coinvariant Modules Conjecture

“…The coefficients of each s µ (z), in the Schur expansion of DBF n (k; j; z), is a polynomial in k and j, with coefficients in Q. Hence, this is also the case for the associated dimension 9 DBF n (k; j).…”

“…Much interesting work has been done recently on diagonal coinvariant spaces spaces in both commuting and anticommuting variables. See for instance [9,10,11,15]. The purpose of this short note is to present a general conjecture expressing the fact that one can simply calculate all cases of multivariate diagonal coinvariant modules in k sets of n commuting variables (bosons), and j sets of n anticommuting variables (fermions), just from the generic case of multivariate diagonal coinvariant spaces.…”

Howe duality of the symmetric group and a multiset partition algebra

“…Fix a positive integer k and a partition λ of n. To determine if the multiplicity of W λ CSn in P r (V n,k ) is non-zero for a given r (or equivalently, to determine if there is a module n) ) we are asking if the coefficient of q r is non-zero in s λ 1/(1 − q) k . Let a λ r,k be equal to the number of multiset tableaux of shape λ with r entries from 1 to k. We know by Theorem 3.1 in [OZ4] that this is the multiplicity of W λ CSn . There is an injection of the tableaux enumerated by a λ r,k into the tableaux enumerated by a λ r+1,k by adding an extra entry k in the highest corner of the tableau.…”

“…By Theorem 3.1 of [OZ4], if n be a positive integer and λ ∈ Par n , then the dimension of the irreducible A r,k (n)-module indexed by λ (assuming that it exists) is equal to the number of multiset valued tableaux of shape λ with r values from 1 through k (where a multiset tableaux is a column strict tableaux whose entries are multisets ordered by a total order).…”

Harmonic differential forms for pseudo-reflection groups I. Semi-invariants

“…Recently there has been a great deal of research activity in algebraic combinatorics studying diagonal actions of the symmetric group S n on k sets of n commuting indeterminants and ℓ sets of n anti-commuting indeterminants. Orellana-Zabrocki [13] describe the S n -invariants of these polynomial rings combinatorially and summarize some of their history. The k = 2, ℓ = 0 case has received a very large amount of attention through the study of the diagonal coinvariants Q[x n , y n ]/D n where D n is the diagonal coinvariant ideal generated by all homogeneous S ninvariants of positive degree and x n is shorthand for x 1 , .…”