Schur indices of perfect groups

It has been noticed by many authors that the Schur indices of the irreducible characters of many quasi-simple finite groups are at most . A conjecture has emerged that the Schur indices of all irreducible characters of all quasi-simple finite groups are at most . We prove that this conjecture cannot be extended to the set of all finite perfect groups. Indeed, we prove that, given any positive integer , there exist irreducible characters of finite perfect groups of chief length which have Schur index .

     

Schur indices of association schemes

In this paper we introduce and investigate the theory of Schur indices arising from simple components of the rational adjacency algebras of association schemes and investigate methods for computing these indices.     

Using character correspondences for Schur index computations

We present a new method of computing the Schur index associated with an irreducible complex character of a finite group over the field of rational numbers. The method uses the Isaacs–Dade character correspondence to reduce to a subgroup of the original group.     

Schur indices and abelian defect groups

We show that the irreducible characters lying in 2-blocks of finite solvable groups with abelian defect groups have trivial 2-local Schur indices.     

Characters, fields, Schur indices and divisibility

Let G be a finite nilpotent group. Suppose that G ₀ is a subgroup of G and that ${\psi}$ is an irreducible character of G ₀. Consider the set S whose elements are the natural numbers $${\rm m}_{\bf Q}(\chi)[{\bf Q}(\chi) : {\bf Q}]$$ as ${\chi}$ runs through the irreducible characters of G which contain ${\psi}$ as a summand when restricted to G ₀. Here m _Q (χ) is, as usual, the rational Schur index of ${\chi}$ , and ${[{\bf Q}(\chi) : {\bf Q}]}$ is the degree of the extension of the field of values of the character as an extension of the rationals. We prove that then the minimum element of S divides all the other elements of S. The result is not true when G is an arbitrary finite group. We also consider some variations of this result.     

Schur indices of characters of groups related to finite sporadic simple groups

LetG be a finite sporadic simple group. Then there exist groupsn.G., n.G.2 and, in casen is even,n.G.2i, the group isoclinic to but not isomorphic ton.G.2. The Schur indices of all irreducible characters of these groups are computed. In a previous paper this was done for the groupsn.G (with one exception). The division algebra corresponding to a character is determined by all the local Schur indices. These are all listed in the tables in Section 6 using the notation from the ATLAS.     

The Schur algorithm and coefficient characterizations for generalized Schur functions

In this paper we analyze the existence of a Schur algorithm and obtain coefficient characterizations for the functions in a generalized Schur class. An application to an interpolation problem of Carathéodory type raised by M.G. Krein and H. Langer is indicated.

     

The first and second Zagreb indices of some graph operations

In this paper some exact expressions for the first and second Zagreb indices of graph operations containing the Cartesian product, composition, join, disjunction and symmetric difference of graphs will be presented. We apply some of our results to compute the Zagreb indices of arbitrary C₄ tube, C₄ torus and q-multi-walled polyhex nanotorus.     

On some properties of factorization indices

Various kinds of factorization indices are considered (right partial indices, left partial indices, Birkhoff indices), and some connections between them are described. We solve also the problem on the relation between the partial indices of two matrix functions and of their product.Dedicated to the memory of Mark Grigorievich KreinThis research was supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

Schur dynamics of the Schur processes

We construct discrete time Markov chains that preserve the class of Schur processes on partitions and signatures.One application is a simple exact sampling algorithm for qvolume-distributed skew plane partitions with an arbitrary back wall. Another application is a construction of Markov chains on infinite Gelfand–Tsetlin schemes that represent deterministic flows on the space of extreme characters of the infinite-dimensional unitary group.     

Diagonal dominance, Schur complements and some classes of H-matrices and P-matrices

We analyze a class of matrices generalizing strictly diagonally dominant matrices and included in the important class of H-matrices. Adequate pivoting strategies and the corresponding Schur complements are studied. A new class of matrices with all their principal minors positive is presented.     

The Flagged Double Schur Function

The double Schur function is a natural generalization of the factorial Schur function introduced by Biedenharn and Louck. It also arises as the symmetric double Schubert polynomial corresponding to a class of permutations called Grassmannian permutations introduced by A. Lascoux. We present a lattice path interpretation of the double Schur function based on a flagged determinantal definition, which readily leads to a tableau interpretation similar to the original tableau definition of the factorial Schur function. The main result of this paper is a combinatorial treatment of the flagged double Schur function in terms of the lattice path interpretations of divided difference operators. Finally, we find lattice path representations of formulas for the symplectic and orthogonal characters for sp(2n) and so(2n + 1) based on the tableau representations due to King and El-Shakaway, and Sundaram. Based on the lattice path interpretations, we obtain flagged determinantal formulas for these characters.     

Equality of Schur and Skew Schur Functions

We determine the precise conditions under which any skew Schur function is equal to a Schur function over both infinitely and finitely many variables. Received May 29, 2004     

The Schur Lie-multiplier of Leibniz algebras

Abstract

For a free presentation 0 → τ → → → 0 of a Leibniz algebra , the Baer invariant is called the Schur multiplier of relative to the Liezation functor or Schur Lie-multiplier. For a two-sided ideal of a Leibniz algebra , we construct a four-term exact sequence relating the Schur Lie-multipliers of and /, which is applied to study and characterize Lie-nilpotency, Lie-stem covers and Lie-capability of Leibniz algebras.     

The Number of Monochromatic Schur Triples

In this paper, we prove that in every 2-coloring of the set {1, , N } = R B, one can find at least N² / 22 + O(N) monochromatic solutions of the equation x + y = z. This solves a problem of Graham et al. .