Let W be a finite Coxeter group and X a subset of W . The length polynomial LW,X(t) is defined by LW,X(t) = ∑ x∈X t `(x), where ` is the length function on W . In this article we derive expressions for the length polynomial where X is any conjugacy class of involutions, or the set of all involutions, in any finite Coxeter group W . In particular, these results correct errors in [6] for the invo...

We prove that the grades of simple modules indexed by boolean permutations, over incidence algebra symmetric group with respect to Bruhat order, are given Lusztig's $\mathbf{a}$-function. Our arguments combinatorial, and include a description intersection two principal order ideals when at least one permutation is boolean. An important object in our work reduced word written as minimally many r...

In this paper, we study a combinatorial problem originating in the following conjecture of Erdős and Lemke: given any sequence of n divisors of n, repetitions being allowed, there exists a subsequence the elements of which are summing to n. This conjecture was proved by Kleitman and Lemke, who then extended the original question to a problem on a zero-sum invariant in the framework of finite Ab...

Given a division ring K containing the field k in its center and two finite subsets A and B of K∗, we give some analogues of Plünnecke and Kneser Theorems for the dimension of the k-linear span of the Minkowski product AB in terms of the dimensions of the k-linear spans of A and B. We also explain how they imply the corresponding more classical theorems for abelian groups. These Plünnecke type ...

For any reduced decomposition i = (i1, i2, . . . , iN ) of a permutation w and any ring R we construct a bijection Pi : (x1, x2, . . . , xN ) 7→ Pi1(x1)Pi2(x2) · · · PiN (xN ) from R to the Schubert cell of w, where Pi1(x1), Pi2(x2), . . . , PiN (xN ) stand for certain elementary matrices satisfying Coxeter-type relations. We show how to factor explicitly any element of a Schubert cell into a p...

We study weight posets of weight multiplicity free (WMF) representations of reductive Lie algebras. Specifically, we are interested in relations between dimR and the number of edges in the Hasse diagram of the corresponding weight poset #E(R). We compute the number of edges and upper covering polynomials for the weight posets of all WMFrepresentations. We also point out non-trivial isomorphisms...

We study the distribution of the number ξn,r of r consecutive records in a sequence of n independent and identically distributed random variables from a common continuous distribution, or equivalently, in a random permutation of n elements. We show that the asymptotic distribution of ξn,r is Poisson for r = 1, 2 and non-Poisson for r ≥ 3. Precise asymptotic results are derived for four probabil...

In this paper, we consider the projective special linear group $PSL_2(59)$ and construct some 1-designs by applying the Key-Moori method on $PSL_2(59)$. Moreover, we obtain parameters of these designs and their automorphism groups. It is shown that $PSL_2(59)$ and $PSL_2(59):2$ appear as the automorphism group of the constructed designs.

Let V be Euclidean space. Let W ⊂ GL(V ) be a finite irreducible reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H ∈ A choose αH ∈ V ∗ such that H = ker(αH). The arrangement A is known to be free: the derivation module D(A) = {θ ∈ DerS | θ(αH ) ∈ SαH} is a free S-module with generators of degrees equal to the exponents of ...

The conjugacy classes of nilpotent n × n matrices can be parametrised by partitions λ of n, and for a nilpotent η in the class parametrised by λ, the variety Fη of η-stable flags has its irreducible components parametrised by the standard Young tableaux of shape λ. We indicate how several algorithmic constructions defined for Young tableaux have significance in this context, thus extending Stei...