Patty Commins - Research

Research

My research interests lie in algebraic combinatorics and representation theory.

More specifically, I study the interplay between the representation theory of algebraic structures -- including finite groups, finite monoids, and finite-dimensional algebras -- and combinatorial structures -- including poset topology, symmetric functions, and hyperplane arrangements. I am also interested in applications of these connections to the study of random walks on symmetric group algebras and their generalizations.

Preprints, Publications, and Slides

Left regular bands with symmetry (with Benjamin Steinberg)
- arXiv preprint
- Slides and a recording of a different version
The q-deformed random-to-random family in the Hecke algebra (with Sarah Brauner, Darij Grinberg, and Franco Saliola)
Spectrum of random-to-random shuffling in the Hecke algebra (with Ilani Axelrod-Freed, Sarah Brauner, Judy Hsin-Hui Chiang, and Veronica Lang)
- arXiv preprint
- Extended abstract (FPSAC 2025 proceedings)
- Slides
Superspace coinvariants and hyperplane arrangements (with Robbie Angarone, Trevor Karn, Satoshi Murai, and Brendon Rhoades)
- Journal article (Advances in Mathematics)
- arXiv preprint
Invariant theory for the face algebra of the braid arrangement
- Journal article (Algebraic Combinatorics)
- arXiv preprint
- Extended abstract (FPSAC 2024 proceedings)
- Slides and a recording
Invariant theory for the free left-regular band and a q-analogue (with Sarah Brauner and Vic Reiner)
- Journal article (Pacific Journal of Mathematics)
- arXiv preprint
- Extended abstract (FPSAC 2023 proceedings)
- Slides and a recording
- Poster
Hexagonal Mosaic Links Generated by Saturation (with John Bush, Tamara Gomez, and Jennifer McLoud-Mann)
- From my time as an undergraduate participant in the University of Washington, Bothell Mathematics REU (in 2017)
- journal article (Journal of Knot Theory and its Ramifications).
Summarizing Diverging String Sequences with Applications to Chain-Letter Petitions (with David Liben-Nowell, Tina Liu, and Kiran Tomlinson)
- Undergraduate Research at Carleton College in Computer Science
- Conference proceedings (CPM 2020).
From my time as an undergraduate participant in the University of Minnesota Combinatorics REU 2018:
- Recovering Conductances of Resistor Networks in a Punctured Disk (with Yulia Alexandr, Naomi Burks, and Sunita Chepuri)
  - arXiv preprint
- Deformations of the Weyl Character Formula for SO(2n + 1) via Ice Models (with Yulia Alexandr, Alexandra Embry, Sylvia Frank, Yutong Li, and Alexander Vetter)
  - arXiv preprint

PhD Thesis

My thesis focuses on how a special class of semigroups called left regular bands behave under group actions.defense.

It covers my paper Invariant theory for the face algebra of the braid arrangement and much of my joint paper Left regular bands with symmetry with Benjamin Steinberg. It also contains quite a bit of exposition on left regular bands which arise from CAT(0)-cube complexes (such as from the space of phylogenetic trees) which did not make it into either of these papers.