arXiv subjects

Sam Jeralds

Publications and source records attributed to Sam Jeralds.

10 recordsLinked to original sources

arXiv2026

Components of $V(m\rho) \otimes V(n\rho)$

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody Lie algebra and let $\rho$ denote the sum of the fundamental weights. The irreducible highest weight representations $V(m\rho)$ occupy a distinguished position in representation theory due to their rich symmetry and geometric significance. In this paper, we study the tensor products \[ V(m\rho)\otimes V(n\rho), \quad m,n \in \mathbb{N}, \] and investigate the structure of their irreducible decompositions. Motivated by the classical conjecture of Kostant, which predicts a highly structured behavior in simpler settings, we propose a general framework describing the irreducible components appearing in such tensor products for finite-dimensional semisimple or affine Kac-Moody Lie algebras $\mathfrak{g}$. Our results identify a family of dominant weights governing the decomposition and provide criteria for their occurrence. This work extends the scope of Kostant-type phenomena and reveals new structural patterns in tensor products associated with multiples of the Weyl vector.

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arXiv2026

Reduction rules for Demazure modules

For $G$ a complex reductive group and $B \subseteq G$ a Borel subgroup, we provide a reduction rule for certain weight multiplicities in Demazure modules $V_\lambda^w$: given a weight $\mu$ on a face of the associated weight polytope $P_\lambda^w$, we reduce the computation of the dimension of the weight space $V_\lambda^w(\mu)$ to a similar problem of computing the weight space dimension for a Demazure module of a Levi subgroup of $G$.

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arXiv2024

Geometric invariant theory and stretched Kostka quasi-polynomials

For $G$ a semisimple, simply-connected complex algebraic group and two dominant integral weights $\lambda, \mu$, we consider the dimensions of weight spaces $V_\lambda(\mu)$ of weight $\mu$ in the irreducible, finite-dimensional highest weight $\lambda$ representation. For natural numbers $N$, the function $N \mapsto \dim V_{N\lambda}(N\mu)$ is a quasi-polynomial in $N$, the stretched Kostka quasi-polynomial. Using methods of geometric invariant theory (GIT), we realize the degree of this quasi-polynomial as the dimension of a certain GIT quotient. As a result, we resolve a conjecture of Gao and Gao on an explicit formula for this degree. We also discuss periods of this quasi-polynomial determined by the GIT approach, and give computational evidence supporting a geometric determination of the minimal period.

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arXiv2024

Irreducible components in Hochschild cohomology of flag varieties

Let $G$ be a simple, simply-connected complex algebraic group with Lie algebra $\mathfrak{g}$, and $G/B$ the associated complete flag variety. The Hochschild cohomology $HH^\bullet(G/B)$ is a geometric invariant of the flag variety related to its generalized deformation theory and has the structure of a $\mathfrak{g}$-module. We study this invariant via representation-theoretic methods; in particular, we give a complete list of irreducible subrepresentations in $HH^\bullet(G/B)$ when $G=SL_n(\mathbb{C})$ or is of exceptional type (and conjecturally for all types) along with nontrivial lower bounds on their multiplicities. These results follow from a conjecture due to Kostant on the structure of the tensor product representation $V(\rho) \otimes V(\rho)$.

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arXiv2024

Affine Demazure Weight Polytopes and Twisted Bruhat Orders

For an untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ with Cartan and Borel subalgebras $\mathfrak{h} \subset \mathfrak{b} \subset \mathfrak{g}$, affine Demazure modules are certain $U(\mathfrak{b})$-submodules of the irreducible highest-weight representations of $\mathfrak{g}$. We introduce here the associated affine Demazure weight polytopes, given by the convex hull of the $\mathfrak{h}$-weights of such a module. Using methods of geometric invariant theory, we determine inequalities which define these polytopes; these inequalities come in three distinct flavors, specified by the standard, opposite, or semi-infinite Bruhat orders. We also give a combinatorial characterization of the vertices of these polytopes lying on an arbitrary face, utilizing the more general class of twisted Bruhat orders.

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arXiv2022

Components of $V(\rho) \otimes V(\rho)$ and dominant weight polyhedra for affine Kac-Moody Lie algebras

Kostant asked the following question: Let $\mathfrak{g}$ be a simple Lie algebra over the complex numbers. Let $\lambda$ be a dominant integral weight. Then, $V(\lambda)$ is a component of $V(\rho)\otimes V(\rho)$ if and only if $\lambda \leq 2 \rho$ under the usual Bruhat-Chevalley order on the set of weights. In an earlier work with R. Chirivi and A. Maffei the second author gave an affirmative answer to this question up to a saturation factor. The aim of the current work is to extend this result to untwisted affine Kac-Moody Lie algebra $\mathfrak{g}$ associated to any simple Lie algebra $\mathring{\mathfrak{g}}$ (up to a saturation factor). In fact, we prove the result for affine $sl_n$ without any saturation factor. Our proof requires some additional techniques including the Goddard-Kent-Olive construction and study of the characteristic cone of non-compact polyhedra.

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arXiv2022

Weight polytopes and saturation of Demazure characters

For $G$ a reductive group and $T\subset B$ a maximal torus and Borel subgroup, Demazure modules are certain $B$-submodules, indexed by elements of the Weyl group, of the finite irreducible representations of $G$. In order to describe the $T$-weight spaces that appear in a Demazure module, we study the convex hull of these weights - the Demazure polytope. We characterize these polytopes both by vertices and by inequalities, and we use these results to prove that Demazure characters are saturated, in the case that $G$ is simple of classical Lie type. Specializing to $G=GL_n$, we recover results of Fink, M\'esz\'aros, and St. Dizier, and separately Fan and Guo, on key polynomials, originally conjectured by Monical, Tokcan, and Yong.

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arXiv2021

Toward a Jacobson--Morozov theorem for Kac--Moody Lie algebras

For a finite-dimensional semisimple Lie algebra $\mathfrak{g}$, the Jacobson--Morozov theorem gives a construction of subalgebras $\mathfrak{sl}_2 \subset \mathfrak{g}$ corresponding to nilpotent elements of $\mathfrak{g}$. In this note, we propose an extension of the Jacobson--Morozov theorem to the symmetrizable Kac--Moody setting and give a proof of this generalization in the case of rank two hyperbolic Kac--Moody algebras. We also give a proof for an arbitrary symmetrizable Kac--Moody algebra under some stronger restrictions.

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arXiv2021

Vertices of Intersection Polytopes and Rays of Generalized Kostka Cones

Let $\mathscr{K}(G)$ be the rational cone generated by pairs $(λ, μ)$ where $λ$ and $μ$ are dominant integral weights and $μ$ is a nontrivial weight space in the representation $V_λ$ of $G$. We produce all extremal rays of $\mathscr{K}(G)$ by considering the vertices of corresponding intersection polytopes $IP_λ$, the set of points in $\mathscr{K}(G)$ with first coordinate $λ$. We show that vertices of $IP_{\varpi_i}$ arise as lifts of vertices coming from cones $\mathscr{K}(L)$ associated to simple Levi subgroups possessing the simple root $α_i$. As corollaries we obtain a complete description of all extremal rays, as well as polynomial formulas describing the numbers of extremal rays depending on type and rank.

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arXiv2020

Multiplicity in root components via Geometric Satake

In this note we explicitly construct top-dimensional components of the cyclic convolution varieties. These components correspond (via the geometric Satake equivalence) to irreducible summands $V(λ+μ-Nβ) \subset V(λ) \otimes V(μ)$ for $G^\vee=SL_{n+1}$, where $N\ge 1$ and $β$ is a positive root. Furthermore, we deduce from these constructions a nontrivial lower bound on the multiplicity of these subrepresentations when $β$ is not a simple root. Finally, we demonstrate that not all such top-dimensional components can be realized as closures of orbits.

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