Colored hook formula for a generalized Young diagram
type: Item
instance of: scholarly article
title: Colored hook formula for a generalized Young diagram
zbMATH Open document ID: 1204.05099
author: Kento Nakada
published in: Osaka Journal of Mathematics
publication date: 2009-02-20T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1227708835
review text: The hook formula of Frame-Robinson-Thrall for a classical Young diagram is well known: \[ {\#\text{STab}(Y_{\lambda})=\frac{d!}{\prod_{v\in Y_{\lambda}}h_v}}, \] where \(\text{STab}(Y_{\lambda})\) is the set of standard tableaux of shape \(\lambda\) and \(h_v\) is the hooklength at a cell \(v\) of \(Y_{\lambda}\). The author proves a \textit{coloured hook formula} for a \textit{generalized Young diagram}, which is defined as follows: Let \(\Pi=\{\alpha_i|i\in I\}\) be a set of simple roots of a Kac-Moody algebra \(\mathfrak g\), \(\Phi_+\) the set of real positive roots. An integral weight \(\lambda\) is \textit{pre-dominant} if \(\langle\lambda,\beta^{\vee}\rangle\geq-1\) for all \(\beta\in\Phi_+\). The set of pre-dominant integral weights is denoted \(P_{\geq-1}\). For \(\lambda\in P_{\geq-1}\), \(\text{Path}(\lambda)\) denotes a certain set of sequences in \(\Phi_+\), and we define \(D(\lambda)=\{\beta\in\Phi_+|\langle\lambda,\beta^{\vee}\rangle=-1\}\), called a \textit{diagram of \(\lambda\)}. The main result is the coloured hook formula (Theorem 7.1): If \(\lambda\in P_{\geq-1}\) is finite (i.e., \(D(\lambda)\) is finite), then \[ \sum_{\substack{(\beta_1,\dots,\beta_l)\in\text{Path}(\lambda)\\l\geq 0}}\frac{1}{\beta_1}\frac{1}{\beta_1+\beta_2}\cdots\frac{1}{\beta_1+\cdots+\beta_l}=\prod_{\beta\in D(\lambda)}(1+\frac{1}{\beta}). \] Here both sides are considered as rational functions in the \(\alpha_i\), called \textit{colour variables}. This specializes to the classical hook formula as follows: Taking the lowest degree part, we first obtain Corollary 7.2: \[ \sum_{(\alpha_{i_1},\dots,\alpha_{i_d})\in\text{MPath}(\lambda)}\frac{1}{\alpha_{i_1}}\frac{1}{\alpha_{i_1}+\alpha_{i_2}}\cdots\frac{1}{\alpha_{i_1}+\cdots+\alpha_{i_d}}=\prod_{\beta\in D(\lambda)}\frac{1}{\beta}, \] where \(\text{MPath}(\lambda)\) denotes the set of elements of maximal length in \(\text{Path}(\lambda)\). If we specialize \(\alpha_i\rightarrow 1\) (\(i\in I\)), we obtain Corollary 7.3: \[ …
reviewed by: M. Rafiq Omar
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 17B67
zbMATH DE Number: 5509697
zbMATH Keywords: Young diagram
zbMATH Keywords: hook formula
zbMATH Keywords: Kac-Moody algebra
zbMATH Keywords: pre-dominant integral weight
zbMATH Keywords: generalized Young diagram
zbMATH Keywords: coloured hook formula
zbMATH Keywords: minuscule element
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5509697