Remarks on Schur's conjecture

Let P be a set of n>d points in Rd for d≥2. It was conjectured by Zvi Schur that the maximum number of (d-1)-dimensional regular simplices of edge length diam(P), whose every vertex belongs to P, is n. We prove this statement under the condition that any two of the simplices share at least d-2 vertices. It is left as an open question to decide whether this condition is always satisfied. We also establish upper bounds on the number of all 2- and 3-dimensional simplices induced by a set P⊂R3 of n points which satisfy the condition that the lengths of their sides belong to the set of k largest distances determined by P

Prehomogeneous vector spaces and ergodic theory III

Let H_1=SL(5), H_2=SL(3), H=H_1 \times H_2. It is known that (G,V) is a prehomogeneous vector space (see [22], [26], [25], for the definition of prehomogeneous vector spaces). A non-constant polynomial \delta(x) on V is called a relative invariant polynomial if there exists a character \chi such that \delta(gx)=\chi(g)\delta(x). Such \delta(x) exists for our case and is essentially unique. So we define V^{ss}={x in V such that \delta(x) is not equal to 0}. For x in V_R^{ss}, let H_{x R+}^0 be the connected component of 1 in classical topology of the stabilizer H_{x R}. We will prove that if x in V_R^ss is "sufficiently irrational", H_{x R+}^0 H_Z is dense in H_R

Permutation polynomials and systems of permutation polynomials in several variables over finite rings

This paper will present the historical development of theorems regarding permutation polynomials in several variables over finite fields. Single variable permutation polynomials will be discussed since they are so important to the discussions which will follow. Theorems involving permutation polynomials and systems of permutation polynomials will also be considered. It will be shown that many of the interesting results obtained for finite fields can be generalized to finite rings

A constant term identity featuring the ubiquitous (and mysterious) Andrews-Mills-Robbins-Rumsey numbers 1, 2, 7, 42, 429, …

AbstractAndrews's recent proof of the Mills-Robbins-Rumsey conjectured formula for the number of totally symmetric self-complementary plane partitions is used to derive a new multi-variate constant term identity, reminiscent of, but not implied by, Macdonald's BCn-Dyson identity. The method of proof consists in translating to the language of constant terms an expression of Doran for the desired number in terms of sums of minors of a certain matrix. The question of a direct proof of the identity, which would furnish an alternative proof of the Mills-Robbins-Rumsey conjecture, is raised, and a prize is offered for its solution

Spin Kostka polynomials and vertex operators

An algebraic iterative formula for the spin Kostka-Foulkes polynomial $K^-_{\xi\mu}(t)$ is given using vertex operator realizations of Hall-Littlewood symmetric functions and Schur's Q-functions. Based on the operational formula, more favorable properties are obtained parallel to the Kostka polynomial. In particular, we obtain some formulae for the number of (unshifted) marked tableaux. As an application, we confirmed a conjecture of Aokage on the expansion of the Schur $P$-function in terms of Schur functions. Tables of $K^-_{\xi\mu}(t)$ for $|\xi|\leq6$ are listed.Comment: 19 pages, 5 tables (correction of authors' names