The sandpile group K(G) of a connected graph G, defined to be the torsion part of the cokernel of its Laplacian matrix L(G), is a subtle graph isomorphism invariant with combinatorial, algebraic, and geometric descriptions. Past work has focused on determining the sandpile group of the hypercube. In this project, we study the sandpile group for a more general collection of graphs, the Cayley graphs of the group F r 2 . While the Sylow-p component of such groups has been classified for p = 2, much less is known about the Sylow-2 component. In this paper, we use ring theory to prove a sharp upper bound for the largest Sylow-2 subgroup of these sandpile groups. In the case of the hypercube, we provide an exact formula for the largest n − 1 among its Sylow-2 cyclic factors. We also find the number of Sylow-2 cyclic factors for "generic" Cayley graphs. With these methods, we also classify the sandpile group for r = 2 and r = 3 in the "generic" case.

For an arbitrary finite monoid M and subgroup K of the unit group of M, we prove that there is a bijection between irreducible representations of M with nontrivial K-fixed space and irreducible representations of H K , the convolution algebra of K × K-invariant functions from M to F, where F is a field of characteristic not dividing |K|. When M is reductive and K = B is a Borel subgroup of the group of units, this indirectly provides a connection between irreducible representations of M and those of F[R], where R is the Renner monoid of M. We conclude with a quick proof of Frobenius Reciprocity for monoids for reference in future papers.

We investigate the asymptotic expansion and the renormalized volume of minimal submanifolds of arbitrary codimension in H n+1 . In particular, we derive formulae for the first and second variations of renormalized volume for Y m ⊆ H n+1 when m < n + 1. We apply our formulae to codimension 1 case, exhibiting a small correction to [2] when n = 2. Furthermore, we prove the existence of an asymptotic graphical description of our minimal submanifold, Y , over the boundary cylinder ∂Y ×R + , and we further derive an L 2 -inner-product relationship between u2 and um+1.

This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e. the Cartan type [Formula: see text] Renner monoid) was computed earlier by Dieng et al. [2], using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type [Formula: see text] and use them to obtain an explicit description of the character table for the type [Formula: see text] Renner monoid Hecke algebra.