Alternating subgroups of Coxeter groups
We study combinatorial properties of the alternating subgroup of a Coxeter group, using a presentation of it due to Bourbaki.
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arXiv subjects
Yuval Roichman
Publications and source records attributed to Yuval Roichman.
A Construction of Coxeter Group Representations (II)
An axiomatic approach to the representation theory of Coxeter groups and their Hecke algebras was presented in [1]. Combinatorial aspects of this construction are studied in this paper. In particular, the symmetric group case is investigated in detail. The resulting representations are completely classified and include the irreducible ones.
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On Degrees in the Hasse Diagram of the Strong Bruhat Order
For a permutation $π$ in the symmetric group $S_n$ let the {\it total degree} be its valency in the Hasse diagram of the strong Bruhat order on $S_n$, and let the {\it down degree} be the number of permutations which are covered by $π$ in the strong Bruhat order. The maxima of the total degree and the down degree and their values at a random permutation are computed. Proofs involve variants of a classical theorem of Turán from extremal graph theory.
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A Unified Construction of Coxeter Group Representations (I)
An elementary approach to the construction of Coxeter group representations is presented.
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Statistics on Wreath Products and Generalized Binomial-Stirling Numbers
Various statistics on wreath products are defined via canonical words, "colored" right to left minima and "colored" descents. It is shown that refined counts with respect to these statistics have nice recurrence formulas of binomial-Stirling type. These extended Stirling numbers determine (via matrix inversion) dual systems, which are also shown to have combinatorial realizations within the wreath product. The above setting also gives rise to MacMahon type equi-distribution theorem over subsets with prescribed statistics.
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Signed Mahonians
A classical result of MacMahon gives a simple product formula for the generating function of major index over the symmetric group. A similar factorial-type product formula for the generating function of major index together with sign was given by Gessel and Simion. Several extensions are given in this paper, including a recurrence formula, a specialization at roots of unity and type $B$ analogues.
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Equidistribution and Sign-Balance on 321-Avoiding Permutations
Let $T_n$ be the set of 321-avoiding permutations of order $n$. Two properties of $T_n$ are proved: (1) The {\em last descent} and {\em last index minus one} statistics are equidistributed over $T_n$, and also over subsets of permutations whose inverse has an (almost) prescribed descent set. An analogous result holds for Dyck paths. (2) The sign-and-last-descent enumerators for $T_{2n}$ and $T_{2n+1}$ are essentially equal to the last-descent enumerator for $T_n$. The proofs use a recursion formula for an appropriate multivariate generating function.
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Descent Representations and Multivariate Statistics
Combinatorial identities on Weyl groups of types $A$ and $B$ are derived from special bases of the corresponding coinvariant algebras. Using the Garsia-Stanton descent basis of the coinvariant algebra of type $A$ we give a new construction of the Solomon descent representations. An extension of the descent basis to type $B$, using new multivariate statistics on the group, yields a refinement of the descent representations. These constructions are then applied to refine well-known decomposition rules of the coinvariant algebra and to generalize various identities.
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q Statistics on $S_n$ and Pattern Avoidance
Natural q analogues of classical statistics on the symmetric groups $S_n$ are introduced; parameters like: the q-length, the q-inversion number, the q-descent number and the q-major index. MacMahon's theorem about the equi-distribution of the inversion number and the reverse major index is generalized to all positive integers q. It is also shown that the q-inversion number and the q-reverse major index are equi-distributed over subsets of permutations avoiding certain patterns. Natural q analogues of the Bell and the Stirling numbers are related to these q statistics -- through the counting of the above pattern-avoiding permutations.
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Permutation Statistics on the Alternating Group
Let $A_n\subseteq S_n$ denote the alternating and the symmetric groups on $1,...,n$. MacMahaon's theorem, about the equi-distribution of the length and the major indices in $S_n$, has received far reaching refinements and generalizations, by Foata, Carlitz, Foata-Schutzenberger, Garsia-Gessel and followers. Our main goal is to find analogous statistics and identities for the alternating group $A_{n}$. A new statistic for $S_n$, {\it the delent number}, is introduced. This new statistic is involved with new $S_n$ equi-distribution identities, refining some of the results of Foata-Schutzenberger and Garsia-Gessel. By a certain covering map $f:A_{n+1}\to S_n$, such $S_n$ identities are `lifted' to $A_{n+1}$, yielding the corresponding $A_{n+1}$ equi-distribution identities.
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Hook Interpolations
The hook components of $V^{\otimes n}$ interpolate between the symmetric power $\sym^n(V)$ and the exterior power $\wedge^n(V)$. When $V$ is the vector space of $k\times m$ matrices over $\bbc$, we decompose the hook components into irreducible $GL_k(\bbc)\times GL_m(\bbc)$-modules. In particular, classical theorems are proved as boundary cases. For the algebra of square matrices over $\bbc$, a bivariate interpolation is presented and studied.
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Shape Avoiding Permutations
Permutations avoiding all patterns of a given shape (in the sense of Robinson-Schensted-Knuth) are considered. We show that the shapes of all such permutations are contained in a suitable thick hook, and deduce an exponential growth rate for their number.
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Descent Numbers and Major Indices for the Hyperoctahedral Group
We introduce and study three new statistics on the hyperoctahedral group $B_{n}$, and show that they give two generalizations of Carlitz's identity for the descent number and major index over $S_{n}$. This answers a question posed by Foata.
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The Flag Major Index and Group Actions on Polynomial Rings
A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type $B$, it is equidistributed with length. For more general wreath products it appears in an explicit formula for the Hilbert series of the (diagonal action) invariant algebra.
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Descent Functions and Random Young Tableaux
The expectation of the descent number of a random Young tableau of a fixed shape is given, and concentration around the mean is shown. This result is generalized to the major index and to other descent functions. The proof combines probabilistic arguments together with combinatorial character theory. Connections with Hecke algebras are mentioned.
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Hecke Algebra Actions on the Coinvariant Algebra
Two actions of the Hecke algebra of type A on the corresponding polynomial ring are studied. Both are deformations of the natural action of the symmetric group on polynomials, and keep symmetric functions invariant. We give an explicit description of these actions, and deduce a combinatorial formula for the resulting graded characters on the coinvariant algebra.
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On Characters of Weyl Groups
In this note a combinatorial formula related to the symmetric group is generalized to an arbitrary finite Weyl group.
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