Odd and Even Major Indices and One-Dimensional Characters for Classical Weyl Groups
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Annals of Combinatorics
Odd and Even Major Indices and One-Dimensional Characters for Classical Weyl Groups Francesco Brenti
and Paolo Sentinelli
Abstract. We define and study odd and even analogues of the major index statistics for the classical Weyl groups. More precisely, we show that the generating functions of these statistics, twisted by the one-dimensional characters of the corresponding groups, always factor in an explicit way. In particular, we obtain odd and even analogues of Carlitz’s identity, of the Gessel–Simion Theorem, and a parabolic extension, and refinement, of a result of Wachs. Mathematics Subject Classification. Primary 05A15; Secondary 05E15, 20F55. Keywords. Permutation, Major index, Weyl group, Generating function, One-dimensional character.
1. Introduction In recent years, a new statistic on the symmetric groups has been introduced and studied in relation with vector spaces over finite fields equipped with a certain quadratic form [21]. This statistic combines combinatorial and parity conditions and is now known as the odd inversion number, or odd length [10,12]. Analogous statistics have later been defined and studied for the hyperoctahedral and even hyperoctahedral groups [11,30,31], and more recently for all Weyl groups [12]. A crucial property of this new statistic is that its signed (by length) generating function over the corresponding Weyl group always factors explicitly [12,33]. Another line of research in the last 20 years has been the definition and study of analogues of the major index statistic for the other classical Weyl groups, namely for the hyperoctahedral and even hyperoctahedral groups (see, e.g., [1,5,7,14–16,24], [25,32]) and for finite Coxeter groups [27]. It is now generally recognized that, among these, the ones with the best properties are
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F. Brenti, P. Sentinelli
those first defined by Adin and Roichman in [1] for the hyperoctahedral group and by Biagioli and Caselli in [7] for the even hyperoctahedral group. Our purpose in this work is to define odd (and even) analogues of these major index statistics for the classical Weyl groups and show that their generating function twisted by the one-dimensional characters of the corresponding Weyl group always factors in an explicit way. More precisely, we show that certain multivariate refinements of these generating functions always factor explicitly. As consequences of our results we obtain odd and even analogues of Carlitz’s identity [13], which involves overpartitions, of the Gessel–Simion Theorem (see, e.g., [2, Theorem 1.3]), and of several other results appearing in the literature ([2, Theorems 5.1, 6.1, 6.2] and [6, Theorem 4.8]). We also obtain an extension, and refinement, of a result of Wachs ([34]). The organization of the paper is as follows. In the next section, we recall some definitions and results that are used in the sequel. In Sect. 3, we define and study odd and even analogues of the major index and descent statistics of the symmetric group (Definition 3.1). In particular, we obtain odd and even analogues
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