arXiv subjects

Alexander Vetter

Publications and source records attributed to Alexander Vetter.

4 recordsLinked to original sources

arXiv2020

Characters of Renner Monoids and Their Hecke Algebras

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 $A$ Renner monoid) was computed earlier by Dieng, Halverson, and Poladian 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 $C$ and use them to obtain an explicit description of the character table for the type $C$ Renner monoid Hecke algebra.

math.RT↗

arXiv2018

Deformations of the Weyl Character Formula for $SO(2n+1,\mathbb{C})$ via Ice Models

We explore combinatorial formulas for deformations of highest weight characters of the odd orthogonal group $SO(2n+1)$. Our goal is to represent these deformations of characters as partition functions of statistical mechanical models -- in particular, two-dimensional solvable lattice models. In Cartan type $A$, Hamel and King [8] and Brubaker, Bump, and Friedberg [3] gave square ice models on a rectangular lattice which produced such a deformation. Outside of type $A$, ice-type models were found using rectangular lattices with additional boundary conditions that split into two classes -- those with `nested' and `non-nested bends.' Our results fill a gap in the literature, providing the first such formulas for type $B$ with non-nested bends. In type $B$, there are many known combinatorial parameterizations of highest weight representation basis vectors as catalogued by Proctor [19]. We show that some of these permit ice-type models via appropriate bijections (those of Sundaram [21] and Koike-Terada [15]) while other examples due to Proctor do not.

math.CO↗

arXiv2018

Convolution Algebras for Finite Reductive Monoids

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 $\mathcal{H}_K$, the convolution algebra of $K\times K$-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.

math.RT↗

arXiv2017

Batch Codes from Hamming and Reed-Müller Codes

Batch codes, introduced by Ishai et al. encode a string $x \in Σ^{k}$ into an $m$-tuple of strings, called buckets. In this paper we consider multiset batch codes wherein a set of $t$-users wish to access one bit of information each from the original string. We introduce a concept of optimal batch codes. We first show that binary Hamming codes are optimal batch codes. The main body of this work provides batch properties of Reed-Müller codes. We look at locality and availability properties of first order Reed-Müller codes over any finite field. We then show that binary first order Reed-Müller codes are optimal batch codes when the number of users is 4 and generalize our study to the family of binary Reed-Müller codes which have order less than half their length.

cs.IT↗