Kostant–Kumar模：表示与重数
Kostant--Kumar modules: presentation and multiplicities
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中文总结 AI 辅助
该论文研究可对称化Kac–Moody代数上的Kostant–Kumar模，给出其生成元与关系表示，证明其分解数可通过重数空间商的正交投影计算，还得到分解数上界并研究Schur正性。
中文摘要 AI 辅助
Kostant–Kumar模$K(\lambda,w,\mu)$是可对称化Kac–Moody代数上不可约最高权模的张量积$V(\lambda)\otimes V(\mu)$的子模，由Weyl群元素$w$索引；其分解数$c^\nu_{\lambda\mu}(w)$是普通张量积重数的精细化。我们从模论角度研究它们，证明$c^\nu_{\lambda\mu}(w)$可通过Kostant–Parthasarathy–Ranga Rao–Varadarajan重数空间的自然商，经正交投影到Demazure模计算。对有限维半单李代数或对称Kac–Moody代数$\mathfrak{g}$，我们给出$K(\lambda,w,\mu)$的生成元与关系表示，扩展了Joseph、Polo和Mathieu的Demazure模表示，将该表示应用于获得$c^\nu_{\lambda\mu}(w)$的上界并研究Schur正性。
英文摘要
Kostant--Kumar modules $K(λ,w,μ)$ are submodules of a tensor product $V(λ)\otimes V(μ)$ of irreducible highest weight modules over a symmetrizable Kac--Moody algebra, indexed by Weyl group elements $w$; their decomposition numbers $c^ν_{λμ}(w)$ refine ordinary tensor product multiplicities. We study them module-theoretically. We show that $c^ν_{λμ}(w)$ is computed by a natural quotient of the Kostant--Parthasarathy--Ranga Rao--Varadarajan multiplicity space, via orthogonal projection onto a Demazure module. For $\mathfrak{g}$ finite-dimensional semisimple or symmetric Kac--Moody, we present $K(λ,w,μ)$ by generators and relations, extending the presentation of Demazure modules due to Joseph, Polo and Mathieu. We apply the presentation to obtain upper bounds on $c^ν_{λμ}(w)$ and to study Schur positivity.
