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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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Full Text: Euclid
References:
[1] J.B. Carrell: Vector fields, flag varieties and Schubert calculus ; in Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Manoj Prakashan, Madras, 1991, 23–57. · Zbl 0795.14026
[2] V.G. Kac: Infinite-Dimensional Lie Algebras, third edition, Cambridge Univ. Press, Cambridge, 1990.
[3] N. Kawanaka: Coxeter groups and Nakayama algorithm , in preparation.
[4] R.V. Moody and A. Pianzola: Lie Algebras with Triangular Decompositions, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, New York, 1995. · Zbl 0874.17026
[5] K. Nakada: \(q\)-Hook formula for a generalized Young diagram ,
[6] S. Okamura: An algorithm which generates a random standard tableau on a generalized Young diagram , Master’s thesis, Osaka university (2003), (in Japanese).
[7] S. Okamura: in preparation. · Zbl 0016.16802
[8] R.A. Proctor: Minuscule elements of Weyl groups, the numbers game, and \(d\)-complete posets , J. Algebra 213 (1999), 272–303. · Zbl 0969.05068 · doi:10.1006/jabr.1998.7648
[9] R.A. Proctor: Dynkin diagram classification of \(\lambda\)-minuscule Bruhat lattices and of \(d\)-complete posets , J. Algebraic Combin. 9 (1999), 61–94. · Zbl 0920.06003 · doi:10.1023/A:1018615115006
[10] B.E. Sagan: The Symmetric Group, Representations, Combinatorial Algorithms, and Symmetric Functions, second edition, Springer, New York, 2001. · Zbl 0964.05070
[11] R.P. Stanley: Ordered structures and partitions , Ph.D. thesis, Harvard University (1971).
[12] J.R. Stembridge: Minuscule elements of Weyl groups , J. Algebra 235 (2001), 722–743. · Zbl 0973.17034 · doi:10.1006/jabr.2000.8488
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