Standard posets and integral weight bases for symmetric powers of minuscule representations

TL;DR

This paper develops a combinatorial framework using standard posets to construct explicit integer weight bases for symmetric powers of minuscule representations of Lie algebras, generalizing previous work and providing concrete bases.

Contribution

It introduces the concept of standard posets for symmetric powers of minuscule representations and proves that certain colored posets are standard, enabling explicit basis construction.

Findings

Standard posets are shown to be suitable for basis construction.

Provides a combinatorial description of bases for symmetric powers.

Results are independent of Lie type and algebraic geometry techniques.

Abstract

This paper extends our earlier work where we constructed ``minuscule'' representations of Kac--Moody algebras from colored posets in a way that maintains key properties of the well-known minuscule representations of simple Lie algebras. In this paper we work only with finite posets. We define standard posets here as ones that can be used to construct weight bases of $m^\text{th}$ symmetric powers ($m \ge 1$) of these minuscule Kac--Moody representations over the integers in a certain fashion. Our main result is to show that our ``$\Gamma$-colored $d$-complete'' and ``$\Gamma$-colored minuscule'' posets are standard. When the algebra at hand is a simply laced simple Lie algebra and the representation minuscule in the classic sense (i.e. isomorphic to irreducible $V(\lambda)$ for minuscule highest weight $\lambda$), our result produces a concrete combinatorially described weight basis for the irreducible representation $V(m\lambda)$ that is indexed in a natural fashion by $m$-multichains in the weight lattice for $V(\lambda)$. C.S. Seshadri first showed such an indexing of a basis is possible. Our work here is entirely combinatorial and does not use results or techniques from algebraic geometry. Constructions in this paper are independent of Lie type and actions of Kac--Moody algebra elements on basis vectors are effectively specified.