About

Dr. Karel Casteels received his PhD in 2010 from Simon Fraser University in Canada under the supervision of Jason Bell. He obtained a Visiting Assistant Professorship at UCSB where he researched the connections between quantum matrices and combinatorics. He held a Marie Curie Research Fellowship at the University of Kent, United Kingdom from 2013-2015, during which he expanded his work to examine other quantum algebras. Since 2015, he has been a Lecturer at UCSB, holding a cross-appointment between the Department of Mathematics and the College of Creative Studies. During the summers, he continues his quantum algebra research with undergraduate students through the Research Experience for Undergraduates (REU) program at UCSB.

Research topics

• Mathematics
• Discrete mathematics
• Computer Science
• Combinatorics

Selected publications

• Scaffoldings of totally positive matrices and line insertion

  Linear and Multilinear Algebra · 2022

  1st authorCorresponding
  • Mathematics
  • Combinatorics
  • Discrete mathematics

  Given a totally positive matrix, can one insert a line (row or column) between two given lines while maintaining total positivity? This question was first posed and solved by Johnson and Smith who gave an algorithm that results in one possible line insertion. In this work, we revisit this problem. First, we show that every totally positive matrix can be associated with a certain vertex-weighted graph in such a way that the entries of the matrix are equal to sums over certain paths in this graph. We call this graph a scaffolding of the matrix. We then use this to give a complete characterization of all possible line insertions as the strongly positive solutions to a given homogeneous system of linear equations.

  PublisherDOI

• Scaffoldings of Totally Positive Matrices and Line Insertion

  arXiv (Cornell University) · 2021

  1st authorCorresponding
  • Computer Science
  • Combinatorics
  • Mathematics

  Given a totally positive matrix, can one insert a line (row or column) between two given lines while maintaining total positivity? This question was first posed and solved by Johnson and Smith who gave an algorithm that results in one possible line insertion. In this work we revisit this problem. First we show that every totally positive matrix can be associated to a certain vertex-weighted graph in such a way that the entries of the matrix are equal to sums over certain paths in this graph. We call this graph a scaffolding of the matrix. We then use this to give a complete characterization of all possible line insertions as the strongly positive solutions to a given homogeneous system of linear equations.

  PublisherOA PDFDOI

• From Grassmann Necklaces to Restricted Permutations and Back Again

  Algebras and Representation Theory · 2017-05-08 · 5 citations

  preprintOpen access1st authorCorresponding
  PublisherOA PDFDOI

• QUANTUM MATRICES BY PATHS

  2016-08-17 · 13 citations

  article1st authorCorresponding

  We study, from a combinatorial viewpoint, the quantized coordinate ring of mxn matrices over an infinite field K (also called quantum matrices) and its torus-invariant prime ideals. The first part of this paper shows that this algebra, traditionally defined by generators and relations, can be seen as subalgebra of a quantum torus by using paths in a certain directed graph. Roughly speaking, we view each generator of quantum matrices as a sum over paths in the graph, each path being assigned an element of the quantum torus. The quantum matrices relations then arise naturally by considering intersecting paths. This viewpoint is closely related to Cauchon's deleting-derivations algorithm. The second part of this paper is to apply the paths viewpoint to the theory of torus-invariant prime ideals of quantum matrices. We prove a conjecture of Goodearl and Lenagan that all such prime ideals, when the quantum parameter q is a non-root of unity, have generating sets consisting of quantum minors. Previously, this result was known to hold only for char(K)=0 and q transcendental over Q. Our strategy is to show that the quantum minors in a given torus-invariant ideal form a Grobner basis.

  Publisher

• Enumeration of torus-invariant strata with respect to dimension in the big cell of the quantum minuscule Grassmannian of type 𝐁_{𝐧}

  Contemporary mathematics - American Mathematical Society · 2012-01-01

  other

  The aim of this article is to give explicit formulae for various generating functions, including the generating function of torus-invariant primitive ideals in the big cell of the quantum minuscule grassmannian of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript n"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">B_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .

  PublisherDOI

• Primitive ideals in quantum Schubert cells: Dimension of the strata

  Forum Mathematicum · 2012-01-13 · 10 citations

  preprintOpen access

  Abstract. The aim of this paper is to study the representation theory of quantum Schubert cells. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔤</m:mi> </m:math> $\mathfrak {g}$ be a simple complex Lie algebra. To each element w of the Weyl group W of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔤</m:mi> </m:math> $\mathfrak {g}$ , De Concini, Kac and Procesi have attached a subalgebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mi>q</m:mi> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mi>w</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:math> $U_q[w]$ of the quantised enveloping algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mi>q</m:mi> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>𝔤</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $U_q(\mathfrak {g})$ . Recently, Yakimov showed that these algebras can be interpreted as the (quantum) Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mi>q</m:mi> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mi>w</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:math> $U_q[w]$ . More precisely, it follows from the Stratification Theorem of Goodearl and Letzter, and from recent works of Mériaux–Cauchon and Yakimov, that the primitive spectrum of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mi>q</m:mi> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mi>w</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:math> $U_q[w]$ admits a stratification indexed by those elements <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>v</m:mi> <m:mo>∈</m:mo> <m:mi>W</m:mi> </m:mrow> </m:math> $v \in W$ with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>v</m:mi> <m:mo>≤</m:mo> <m:mi>w</m:mi> </m:mrow> </m:math> $v \le w$ in the Bruhat order. Moreover each stratum is homeomorphic to the spectrum of maximal ideals of a torus. The main result of this paper gives an explicit formula for the dimension of the stratum associated to a pair <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>v</m:mi> <m:mo>≤</m:mo> <m:mi>w</m:mi> </m:mrow> </m:math> $v \le w$ .

  PublisherOA PDFDOI

• Enumeration of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="script">H</mml:mi></mml:math>-strata in quantum matrices with respect to dimension

  Journal of Combinatorial Theory Series A · 2011-08-29 · 6 citations

  articleCorresponding
  PublisherDOI

• Enumeration of torus-invariant strata with respect to dimension in the big cell of the quantum minuscule Grassmannian of type B_n

  arXiv (Cornell University) · 2011-04-14

  preprintOpen access

  The aim of this article is to give explicit formulae for various generating functions, including the generating function of torus-invariant primitive ideals in the big cell of the quantum minuscule grassmannian of type B_n.

  PublisherOA PDFDOI

• A graph theoretic method for determining generating sets of prime ideals in quantum matrices

  Journal of Algebra · 2011-02-06 · 14 citations

  article1st authorCorresponding
  PublisherDOI

• Enumeration of $\C{H}$-strata in quantum matrices with respect to dimension

  arXiv (Cornell University) · 2010-09-13 · 3 citations

  preprintOpen access

  We present a combinatorial method to determine the dimension of $\C{H}$-strata in the algebra of $m\times n$ quantum matrices $\Oq$ as follows. To a given $\C{H}$-stratum we associate a certain permutation via the notion of pipe-dreams. We show that the dimension of the $\C{H}$-stratum is precisely the number of odd cycles in this permutation. Using this result, we are able to give closed formulas for the trivariate generating function that counts the $d$-dimensional $\C{H}$-strata in $\Oq$. Finally, we extract the coefficients of this generating function in order to settle conjectures proposed by the first and third named authors \cite{bldim,bll} regarding the asymptotic proportion of $d$-dimensional $\C{H}$-strata in $\Oq$.

  PublisherOA PDFDOI

Frequent coauthors

• Jason P. Bell

  University of Waterloo

  7 shared

• Stéphane Launois

  University of Kent

  7 shared

• Siân Fryer

  University of California, Santa Barbara

  3 shared

• Brett Stevens

  Carleton University

  1 shared

• R. Bruce Richter

  University of Waterloo

  1 shared

Labs

• Karel Casteels LabPI

Awards & honors

• Marie Curie Research Fellowship at the University of Kent, U…