Colored hook formula for a generalized Young diagram
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: \[ {\#\text{MPath}(\lambda)=\frac{d!}{\prod_{\beta\in D(\lambda)}\text{ht}(\beta)}}. \] Here \(\text{ht}(\beta)\) can be viewed as a hooklength (Theorem 6.8). The following Example is given: In the case of type \(A_3\) and \(\lambda=-\omega_2\), where \(\omega_2\) is the fundamental root corresponding to \(\alpha_2\), we have \(D(\lambda)=\{\alpha_2, \alpha_1+\alpha_2, \alpha_2+\alpha_3, \alpha_1+\alpha_2+\alpha_3\}\), which is a realization of the \(2\times 2\) Young diagram. (In general, Young diagrams are realized as shapes of some \(\lambda\) over root systems of type \(A\).) The author remarks that the \textit{coloured} hook formula obtained is new even for Young diagrams. The main theorem is proved in Sections 8 and 9 by manipulation of roots, weights, Weyl group elements, etc., which are all described in detail. In the last section the author uses Corollary 7.3 to give a new proof of a result of Dale Peterson: \[ {\#\text{Red}(w)=\frac{\ell(w)!}{\prod_{\beta\in \Phi(w)}\text{ht}(\beta)}}, \] where \(w\) is a minuscule element of the Weyl group of \(\mathfrak g\), \(\Phi(w)=\{\beta\in\Phi_+|w^{-1}(\beta)<0\}\), and \(\#\text{Red}(w)\) is the number of reduced decompositions of \(w\).
scientific article; zbMATH DE number 5509697