Colored hook formula for a generalized Young diagram.

*(English)*Zbl 1204.05099The 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 \(w\).

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 \(w\).

Reviewer: M. Rafiq Omar (Bellville)

##### MSC:

05E10 | Combinatorial aspects of representation theory |

17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |

##### Keywords:

Young diagram; hook formula; Kac-Moody algebra; pre-dominant integral weight; generalized Young diagram; coloured hook formula; minuscule element
Full Text:
Euclid

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