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Colored hook formula for a generalized Young diagram. (English) Zbl 1204.05099
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 coloured hook formula for a 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 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 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 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 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 $$w$$.

##### MSC:
 05E10 Combinatorial aspects of representation theory 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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