We will restrict out attention to the simplest possible case, namely G = SL2(R), Γ = SL2(Z), and right K = SO(2)-invariant functions on Γ\G. That is, we neglect the finite primes and holomorphic automorphic forms. Let N be the subgroup of G consisting of upper-triangular unipotent matrices, and P the parabolic subgroup consisting of all upper-triangular matrices. For simplicity we give K total ...

Abstract. Let nn(C) be the algebra of strictly upper-triangular n × n matrices and X2 = {u ∈ nn(C) : u2 = 0} the subset of matrices of nilpotent order 2. Let Bn(C) be the group of invertible upper-triangular matrices acting on nn by conjugation. Let Bu be the orbit of u ∈ X2 with respect to this action. Let S2n be the subset of involutions in the symmetric group Sn. We define a new partial orde...

In studying the enumerative theory of super characters of the group of upper triangular matrices over a finite field we found that the moments (mean, variance and higher moments) of novel statistics on set partitions of [n] = {1, 2, · · · , n} have simple closed expressions as linear combinations of shifted bell numbers. It is shown here that families of other statistics have similar moments. T...

∣ p . Hardy’s inequality thus asserts that the Cesáro matrix operator C = (cj,k), given by cj,k = 1/j, k ≤ j and 0 otherwise, is bounded on lp and has norm ≤ p/(p − 1). (The norm is in fact p/(p − 1).) Hardy’s inequality leads naturally to the study on lp norms of general matrices. For example, we say a matrix A = (aj,k) is a weighted mean matrix if its entries satisfy aj,k = 0, k > j and aj,k ...

Let P(t) denote an infinitely long right triangular prism whose base is an equilateral triangle of edge length t. Let F (t) be the family of those subsets of P(t) that are congruent to a regular tetrahedron of unit edge. We present complete classification of the members of F (t) modulo rigid motions within the prism P(t), for every t > 0.

Let V be an n-dimensional vector space over a finite field Fq and P = {1, 2, . . . , n} a poset. We consider on V the poset-metric dP . In this paper, we give a complete description of groups of linear isometries of the metric space (V, dP ), for any poset-metric dP . We show that a linear isometry induces an automorphism of order in poset P , and consequently we show the existence of a pair of...

A labeled set partition is a partition of a set of integers whose arcs are labeled by nonzero elements of an abelian group A. Inspired by the action of the linear characters of the unitriangular group on its supercharacters, we define a group action of An on the set of A-labeled partitions of an (n + 1)-set. By investigating the orbit decomposition of various families of set partitions under th...

We study a natural random walk over the upper triangular matrices, with entries in the field Z2, generated by steps which add row i + 1 to row i. We show that the mixing time of the lazy random walk is O(n) which is optimal up to constants. Our proof makes key use of the linear structure of the group and extends to walks on the upper triangular matrices over the fields Zq for q prime.