Combinatorics and the Schur algebra

The Rational Schur Algebra

We extend the family of classical Schur algebras in type A, which determine the polynomial representation theory of general linear groups over an infinite field, to a larger family, the rational Schur algebras, which determine the rational representation theory of general linear groups over an infinite field. This makes it possible to study the rational representation theory of such general lin...

THE q-SCHUR ALGEBRA

We study a class of endomomorphism algebras of certain q-permutation modules over the Hecke algebra of type B, whose summands involve both parabolic and quasi-parabolic subgroups, and prove that these algebras are integrally free and quasi-hereditary, and are stable under base change. Some consequences for decomposition numbers are discussed. The notion of a q-Schur algebra was introduced by Di...

Algebra and Combinatorics

The author has a wide variety of topics for projects in Algebra and Combinatorics. Three candidates are an investigation of the graph theoretical properties of the subgroup graph of a group, the study of simplicial monomial rings, and the search for Hurwitz groups. All three projects are appropriate for undergraduates, each has a high probability of success, and each offers a wide range of poss...

Algebra and Infinite Combinatorics

Max-algebra: the linear algebra of combinatorics?

Let a ⊕ b = max(a, b), a ⊗ b = a + b for a, b ∈ R := R ∪ {−∞}. By max-algebra we understand the analogue of linear algebra developed for the pair of operations (⊕,⊗) extended to matrices and vectors. Max-algebra, which has been studied for more than 40 years, is an attractive way of describing a class of nonlinear problems appearing for instance in machinescheduling, information technology and ...