q-Partition algebra combinatorics

q-Partition algebra combinatorics

We study a q-analog Qr(n, q) of the partition algebra Pr(n). The algebra Qr(n, q) arises as the centralizer algebra of the finite general linear group GLn(Fq) acting on a vector space IR q coming from r-iterations of Harish-Chandra restriction and induction. For n ≥ 2r, we show that Qr(n, q) has the same semisimple matrix structure as Pr(n). We compute the dimension dn,r(q) = dim(IR r q ) to be...

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 ...

The rook partition algebra

The rook partition algebra RPk(x) is a semisimple algebra for all but a finite number of values of x ∈ C that arises from looking at what commutes with the action of the symmetric group Sn on U ⊗k, where U is the direct sum of the natural representation and the trivial representation of Sn. We give a combinatorial description of this algebra, construct its irreducible representations, and exhib...