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Nate Harman

Publications and source records attributed to Nate Harman.

At least 19 recordsLinked to original sources
arXiv2025

Representations of finite matrix monoids

Let $\mathfrak{M}_n$ be the multiplicative monoid of $n \times n$ matrices over a finite field. The monoid algebra $\mathbf{C}[\mathfrak{M}_n]$ has been studied for several decades. One of the important early results is Kov\'acs' theorem that the two-sided ideal spanned by matrices of rank at most $r$ has a unit. Our most significant result is an explicit formula for this unit. Prior to our work, such a formula was only known in a few examples. We also study the module theory of $\mathbf{C}[\mathfrak{M}_n]$. We explicitly describe the simple modules, and establish induction and restriction rules. We show that the simple decomposition of an arbitrary module can be determined using character theory of finite general linear groups; this relies on a Pieri rule of Gurevich--Howe. We also establish a version of Schur--Weyl duality for $\mathbf{C}[\mathfrak{M}_n]$. Many of these results hold over more general coefficient fields.

math.RT↗
arXiv2025

Universal properties of Delannoy categories

Recently, the second and third authors introduced a new symmetric tensor category $\underline{\mathrm{Perm}}(G, \mu)$ associated to an oligomorphic group $G$ with a measure $\mu$. When $G$ is the group of order preserving self-bijections of the real line there are four such measures, and the resulting tensor categories are called the Delannoy categories. The first Delannoy category is semi-simple, and was studied in detail by Harman, Snowden, and Snyder. We give universal properties for all four Delannoy categories in terms of ordered \'etale algebras. As a consequence, we show that the second and third Delannoy categories admit at least two local abelian envelopes, and the fourth admits at least four. We also prove a coarser universal property for $\underline{\mathrm{Perm}}(G, \mu)$ for a general oligomorphic group $G$.

math.CT↗
arXiv2025

Kronecker Coefficients, Crystals, and Bitableaux

What might a combinatorial interpretation of the Kronecker coefficients even look like? We introduce a class of combinatorial objects called bitableaux, which we believe are a natural candidate, and we formulate a purely combinatorial problem which if resolved would give a combinatorial interpretation of the Kronecker coefficients. We make some partial progress on this problem -- enough to extract a combinatorial expansion for a Kronecker product of Schur functions in the monomial basis. We also explain how in this framework finding a combinatorial interpretation for Kronecker coefficients can be thought of as looking for a generalization of the RSK and dual RSK insertion algorithms.

math.RT↗
arXiv2025

Classical interpolation categories

We study tensor categories that interpolate the representation categories of finite classical groups. There are (at least) two ways to approach these categories: via ultraproducts and via oligomorphic groups. Both have strengths and weaknesses. The ultraproduct categories are easy to define, but their structure is not clear. On the other hand, the oligomorphic approach requires a certain kind of measure as an input, and the space of measures is not obvious. Furthermore, it is not a priori clear that the two approaches yield the same categories in general. We handle all of these issues: we determine all measures on the oligomorphic groups, and we show that the oligomorphic and ultraproduct categories agree, which gives us basic structural results about the latter. Our results rely upon (and in some sense repackage) enumerative results in finite geometry.

math.RT↗
arXiv2025

The "Shape" of q-Binomial Coefficients

This is an informal and mostly expository note describing some asymptotic behavior and qualitative properties of the q-binomial coefficients. The results are mostly not new, but the overall story we present does not seem to be well known -- and the diagrams are all new.

math.CO↗
arXiv2024

Tensor spaces and the geometry of polynomial representations

A "tensor space" is a vector space equipped with a finite collection of multi-linear forms. In previous work, we showed that (for each signature) there exists a universal homogeneous tensor space, which is unique up to isomorphism. Here we generalize that result: we show that each Zariski class of tensor spaces contains a weakly homogeneous space, which is unique up to isomorphism; here, we say that two tensor spaces are "Zariski equivalent" if they satisfy the same polynomial identities. Our work relies on the theory of $\mathbf{GL}$-varieties developed by Bik, Draisma, Eggermont, and Snowden.

math.RT↗
arXiv2024

Higher Congruences in Character Tables

Motivated by recent work of Peluse and Soundararajan on divisibility properties of the entries of the character tables of symmetric groups, we investigate the question: For a finite group G, when are two columns of the character table of G congruent to one another modulo a power of a prime?

math.RT↗
arXiv2023

Arboreal tensor categories

We introduce some new symmetric tensor categories based on the combinatorics of trees: a discrete family $\mathcal{D}(n)$, for $n \ge 3$ an integer, and a continuous family $\mathcal{C}(t)$, for $t \ne 1$ a complex number. The construction is based on the general oligomorphic theory of Harman--Snowden, but relies on two non-trivial results we establish. The first determines the measures for the class of trees, and the second is a semi-simplicity theorem. These categories have some notable properties: for instance, $\mathcal{C}(t)$ is the first example of a 1-parameter family of pre-Tannakian categories of superexponential growth that cannot be obtained by interpolating categories of moderate growth.

math.RT↗
arXiv2023

Discrete pre-Tannakian categories

Pre-Tannakian categories are a natural class of tensor categories that can be viewed as generalizations of algebraic groups. We define a pre-Tannkian category to be discrete if it is generated by an \'etale commutative algebra; these categories generalize finite groups. The main theorem of this paper establishes a rough classification of these categories: we show that any discrete pre-Tannakian $\mathcal{C}$ category is associated to an oligomorphic group $G$, via a construction we recently introduced. In certain cases, such as when $\mathcal{C}$ has enough projectives, we completely describe $\mathcal{C}$ in terms of $G$.

math.RT↗
arXiv2023

The circular Delannoy category

Let $G$ (resp. $H$) be the group of orientation preserving self-homeomorphisms of the unit circle (resp. real line). In previous work, the first two authors constructed pre-Tannakian categories $\underline{\mathrm{Rep}}(G)$ and $\underline{\mathrm{Rep}}(H)$ associated to these groups. In the predecessor to this paper, we analyzed the category $\underline{\mathrm{Rep}}(H)$ (which we named the ``Delannoy category'') in great detail, and found it to have many special properties. In this paper, we study $\underline{\mathrm{Rep}}(G)$. The primary difference between these two categories is that $\underline{\mathrm{Rep}}(H)$ is semi-simple, while $\underline{\mathrm{Rep}}(G)$ is not; this introduces new complications in the present case. We find that $\underline{\mathrm{Rep}}(G)$ is closely related to the combinatorics of objects we call Delannoy loops, which seem to have not previously been studied.

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arXiv2023

Pre-Galois categories and Fra\"iss\'e's theorem

Galois categories can be viewed as the combinatorial analog of Tannakian categories. We introduce the notion of pre-Galois category, which can be viewed as the combinatorial analog of pre-Tannakian categories. Given an oligomorphic group $G$, the category $\mathbf{S}(G)$ of finitary smooth $G$-sets is pre-Galois. Our main theorem (approximately) says that these examples are exhaustive; this result is, in a sense, a reformulation of Fra\"iss\'e's theorem. We also introduce a more general class of B-categories, and give some examples of B-categories that are not pre-Galois using permutation classes. This work is motivated by certain applications to pre-Tannakian categories.

math.RT↗
arXiv2023

The Delannoy category

Let $G$ be the group of all order-preserving self-maps of the real line. In previous work, the first two authors constructed a pre-Tannakian category $\underline{\mathrm{Rep}}(G)$ associated to $G$. The present paper is a detailed study of this category, which we name the Delannoy category. We classify the simple objects, determine branching rules to open subgroups, and give a combinatorial rule for tensor products. The Delannoy category has some remarkable features: it is semi-simple in all characteristics; all simples have categorical dimension $\pm 1$; and the Adams operations on its Grothendieck group are trivial. We also give a combinatorial model for $\underline{\mathrm{Rep}}(G)$ based on Delannoy paths.

math.RT↗
arXiv2022

Ultrahomogeneous tensor spaces

A cubic space is a vector space equipped with a symmetric trilinear form. Using categorical Fra\"iss\'e theory, we show that there is a universal ultrahomogeneous cubic space $V$ of countable infinite dimension, which is unique up to isomorphism. The automorphism group $G$ of $V$ is quite large and, in some respects, similar to the infinite orthogonal group. We show that $G$ is a linear-oligomorphic group (a class of groups we introduce), and we determine the algebraic representation theory of $G$. We also establish some model-theoretic results about $V$: it is $\omega$-categorical (in a modified sense), and has quantifier elimination (for vectors). Our results are not specific to cubic spaces, and hold for a very general class of tensor spaces; we view these spaces as linear analogs of the relational structures studied in model theory.

math.LO↗
arXiv2022

A Tensor-Cube Version of the Saxl Conjecture

Let $n$ be a positive integer, and let $ρ_n = (n, n-1, n-2, \ldots, 1)$ be the ``staircase'' partition of size $N = {n+1 \choose 2}$. The Saxl conjecture asserts that every irreducible representation $S^λ$ of the symmetric group $S_N$ appears as a subrepresentation of the tensor square $S^{ρ_n} \otimes S^{ρ_n}$. In this short note we show that every irreducible representation of $S_N$ appears in the tensor cube $S^{ρ_n} \otimes S^{ρ_n} \otimes S^{ρ_n}$.

math.RT↗
arXiv2022

Oligomorphic groups and tensor categories

Given an oligomorphic group $G$ and a measure $\mu$ for $G$ (in a sense that we introduce), we define a rigid tensor category $\underline{\mathrm{Perm}}(G; \mu)$ of "permutation modules," and, in certain cases, an abelian envelope $\underline{\mathrm{Rep}}(G; \mu)$ of this category. When $G$ is the infinite symmetric group, this recovers Deligne's interpolation category. Other choices for $G$ lead to fundamentally new tensor categories. For example, we construct the first known semi-simple pre-Tannakian categories in positive characteristic with super-exponential growth. One interesting aspect of our construction is that, unlike previous work in this direction, our categories are concrete: the objects are modules over a ring, and the tensor product receives a universal bi-linear map. Central to our constructions is a novel theory of integration on oligomorphic groups, which could be of more general interest. Classifying the measures on an oligomorphic group appears to be a difficult problem, which we solve in only a few cases.

math.RT↗
arXiv2021

The indecomposable objects in the center of Deligne's category $Rep(S_t)$

We classify the indecomposable objects in the monoidal center of Deligne's interpolation category $Rep(S_t)$ by viewing $Rep(S_t)$ as a model-theoretic limit in rank and characteristic. We further prove that the center of $Rep(S_t)$ is semisimple if and only if $t$ is not a non-negative integer. In addition, we identify the associated graded Grothendieck ring of this monoidal center with that of the graded sum of the centers of representation categories of finite symmetric groups with an induction product. We prove analogous statements for the abelian envelope.

math.RT↗
arXiv2020

$p$-adic dimensions in symmetric tensor categories in characteristic $p$

To every object $X$ of a symmetric tensor category over a field of characteristic $p>0$ we attach $p$-adic integers $\text{Dim}_+(X)$ and $\text{Dim}_-(X)$ whose reduction modulo $p$ is the categorical dimension $\text{dim}(X)$ of $X$, coinciding with the usual dimension when $X$ is a vector space. We study properties of $\text{Dim}_{\pm}(X)$, and in particular show that they don't always coincide with each other, and can take any value in $\mathbb{Z}_p$. We also discuss the connection of $p$-adic dimensions with the theory of $λ$-rings and Brauer characters.

math.RT↗