1,720,992 research outputs found
Closed product formulas for extensions of generalized Verma modules
No full textWe give explicit combinatorial product formulas for the polynomials encoding the dimensions of the spaces of extensions of (g,p)-generalized Verma modules, in the cases when (g,p) corresponds to an indecomposable classic Hermitian symmetric pair. The formulas imply that these dimensions are combinatorial invariants. We also discuss how these polynomials, defined by Shelton, are related to the parabolic R-polynomials introduced by Deodhar
Major and descent statistics for the even-signed permutation group
No full textWe introduce and study three new statistics on the even-signed permutation group D_n. We show that two of these are Mahonian, i.e., are equidistributed with length, and that a pair of them gives a generalization of Carlitz’s identity on the Euler-Mahonian distribution of the descent number and major index over S_n
Signed Mahonian polynomials for classical Weyl groups
No full textAbstractThe generating functions of the major index and of the flag-major index, with each of the one-dimensional characters over the symmetric and hyperoctahedral group, respectively, have simple product formulas. In this paper, we give a factorial-type formula for the generating function of the D-major index with sign over the Weyl groups of type D. This completes a picture which is now known for all the classical Weyl groups
Supersolvable LL-lattices of binary trees
No full textSome posets of binary leaf-labeled trees are shown to be supersolvable lattices and explicit EL-labelings are given. Their characteristic polynomials are computed, recovering their known factorization in a different way. (c) 2005 Elsevier B.V. All rights reserved
Colored-descent representations of complex reflection groups G(r, p, n)
No full textWe study the complex reflection groups G(r, p, n). By considering these groups as subgroups of the wreath products Z(r) integral S-n, and by using Clifford theory, we define combinatorial parameters and descent representations of G(r, p, n), previously known for classical Weyl groups. One of these parameters is the flag major index, which also has an important role in the decomposition of these representations into irreducibles. A Carlitz type identity relating the combinatorial parameters with the degrees of the group, is presented
Enumerating wreath products via Garsia-Gessel bijections
No full textWe generalize two bijections due to Garsia and Gessel to compute the generating functions of the two vector statistics (des(G), maj, l(G), col) and (des(G), ides(G), maj, imaj, col, icol) over the wreath product of a symmetric group by a cyclic group. Here desG, l(G), maj, col, idesG, imaj(G), and icol denote the number of descents, length, major index, color weight, inverse descents, inverse major index, and inverse color weight, respectively. Our main formulas generalize and unify several known identities due to Brenti, Carlitz, Chow-Gessel, Garsia-Gessel, and Reiner on various distributions of statistics over Coxeter groups of type A and B
Tensorial square of the hyperoctahedral group coinvariant space
No full textThe purpose of this paper is to give an explicit description of the trivial and alternating components of the irreducible representation decomposition of the bigraded module obtained as the tensor square of the coinvariant space for hyperoctahedral groups
On some analogous of Carlitz’s identity for the hyperoctahedral group
No full textWe give a new description of the flag major index, introduced by Adin and Roichman, by using a major index defined by Reiner. This allows us to establish a connection between an identity of Reiner and some more recent results due to Chow and Gessel. Furthermore we generalize the main identity of Chow and Gessel by computing the four-variate generating series of descents, major index, length, and number of negative entries over Coxeter groups of type B and D
Combinatorics and representations of complex reflection groups G(r, p, n)
No full textFor every r, n, p|r there is a complex reflection group, denoted G(r, p, n), consisting of all monomial n × n matrices such that all the nonzero entries are r th roots of the unity and the r/p th power of the product of the nonzero entries is 1. By considering these groups as subgroups of the colored permutation groups, Z r wr S n, we use Clifford theory to define on G(r, p, n) combinatorial parameters and descent representations previously defined on Classical Weyl groups. One of these parameters is the major index which also has an important role in the decomposition of descent representations into irreducibles. We present also a Carlitz identity for these complex reflection groups
Some identities involving second kind Stirling numbers of types B and D
Full text linkUsing Reiner’s definition of Stirling numbers of the second kind in types B and D, we generalize two well-known identities concerning the classical Stirling numbers of the second kind. The first identity relates them with Eulerian numbers and the second identity interprets them as entries in a transition matrix between the elements of two standard bases of the polynomial ring R[x]. Finally, we generalize these identities to the group of colored permutations G(m,n)
- …
