Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1206.0404

Sums of squares of the Littlewood-Richardson coefficients and GL(n)-harmonic polynomials

Pamela E. Harris·Jeb F. Willenbring
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the example from invariant theory concerning the conjugation action of the general linear group on several copies of the $n \times n$ matrices, and examine a symmetric function which stably describes the Hilbert series for the invariant ring with respect to the multigradation by degree. The terms of this Hilbert series may be described as a sum of squares of Littlewood-Richardson coefficients. A "principal specialization" of the gradation is then related to the Hilbert series of the $\K$-invariant subring in the $\GL_n$-harmonic polynomials, where $\K$ denotes a block diagonal embedding of a product of general linear groups. We also consider other specializations of this Hilbert series.

Explore related subjects

math.RTmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Pamela E. Harris, Jeb F. Willenbring. 2013-04-22. Sums of squares of the Littlewood-Richardson coefficients and GL(n)-harmonic polynomials. https://arxiv.org/abs/1206.0404

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A local relative trace formula for F*\SL(2,F)

In this note, we derive explicitly the local relative trace formula for the symmetric space F*\SL(2,F) at the level of Lie algebras, where F is a p-adic field of residue characteristic greater than two and F* is the set of invertible elements in F. This is perhaps one of the simplest non-trivial analogs of the trace formula, and also a motivating example for the author's work (in preparation) on the relative trace formula.

math.RT↗
arXiv2026

Remarks on the shuffle elements of Iwahori--Hecke algebras

Let $H_n(q)$ be the generic Iwahori--Hecke algebra associated to the symmetric group $\mathfrak{S}_n$ of degree $n$ and let $\mathscr{Y}_{n,i}$ be the sum of standard basis of $H_n(q)$ with Coxeter length $i$ for $1\leq i\leq \binom{n}{2}$. We show that $\mathscr{Y}_{n,1}$ and $\mathscr{Y}_{n,2}$ are noncommutative whenever $n\geq 4$, which disproves Doikou's Conjecture on the commutativity of the shuffle elements of $H_n(q)$ in (Nuclear Physics B 1029 (2026): 117532). We also show that the matrix of the operator of the left multiplication by $\mathscr{Y}_{n,i}$ in $H_n(q)$ is $q$-symmetric for all $i$.

math.RT↗
arXiv2026

Monoidal categorification on open Richardson varieties

Let $G$ be a simply connected complex semisimple group of finite simply-laced type, and let $v\leq w$ be Weyl group elements. We construct a monoidal categorification of open Richardson varieties in this setting. The construction realizes the cluster variables of Ménard's seed by real simple modules in the quiver Hecke category $\mathscr C_{w,v}$, by comparing their truncated Lusztig parameters with the $Δ$-vectors at every step of Ménard's mutation algorithm. Every mutation in a retained direction is realized inside $\mathscr C_{w,v}$. After localizing at the determinantial modules $M(w\varpi_i,v\varpi_i)$, the resulting category is a monoidal categorification of the coordinate ring of the open Richardson variety $\mathring{\mathcal B}_{v,w}$. In particular, cluster monomials without negative frozen exponents are represented by real simple modules in $\mathscr C_{w,v}$; arbitrary frozen Laurent factors are realized in its localization.

math.RT↗