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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., Comment: 20 pages

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?, Comment: 12 pages. Comments welcome!

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., Comment: 37 pages

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 $G$., Comment: 29 pages

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., Comment: 38 pages

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., Comment: 28 pages

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., Comment: 50 pages

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., Comment: 33 pages

Let $n$ be a positive integer, and let $\rho_n = (n, n-1, n-2, \ldots, 1)$ be the ``staircase'' partition of size $N = {n+1 \choose 2}$. The Saxl conjecture asserts that every irreducible representation $S^\lambda$ of the symmetric group $S_N$ appears as a subrepresentation of the tensor square $S^{\rho_n} \otimes S^{\rho_n}$. In this short note we show that every irreducible representation of $S_N$ appears in the tensor cube $S^{\rho_n} \otimes S^{\rho_n} \otimes S^{\rho_n}$., Comment: 4 pages. Added an independent second proof of the main theorem

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., Comment: 135 pages

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., Comment: v2: Final accepted manuscript after minor edits, to appear in Proceedings of the LMS

We discuss certain effective improvements on superrigidity for $SL_n(\mathbb{Z})$ for finite $n>2$. Using these ideas we then use superrigidity to prove a representation stability theorem about pointwise finite dimensional $VIC(\mathbb{Z})$-modules, which itself can be viewed as a superrigidity theorem for $VIC(\mathbb{Z})$ and $GL_\infty(\mathbb{Z})$, Comment: 40 pages

We classify simple associative and Lie algebras inside the Deligne categories $Rep(S_t)$, answering a question posed by Etingof.

We outline an approach to understanding restrictions of polynomial representations of $GL_n(\mathbb{C})$ to $S_n$ by first restricting to $T \rtimes S_n$, the subgroup of $n \times n$ monomial matrices. Using this approach we give a combinatorial interpretation for the decomposition of a tensor product of symmetric powers of the defining representation., Comment: 24 pages

We define a notion of virtual Specht stability which is a relaxation of the Church-Farb notion of representation stability for sequences of symmetric group representations. Using a structural result of Nagpal, we show that $FI$-modules over fields of positive characteristic exhibit virtual Specht stability., Comment: 10 pages

The partition algebra $\mathsf{P}_k(n)$ and the symmetric group $\mathsf{S}_n$ are in Schur-Weyl duality on the $k$-fold tensor power $\mathsf{M}_n^{\otimes k}$ of the permutation module $\mathsf{M}_n$ of $\mathsf{S}_n$, so there is a surjection $\mathsf{P}_k(n) \to \mathsf{Z}_k(n) := \mathsf{End}_{\mathsf{S}_n}(\mathsf{M}_n^{\otimes k}),$ which is an isomorphism when $n \ge 2k$. We prove a dimension formula for the irreducible modules of the centralizer algebra $\mathsf{Z}_k(n)$ in terms of Stirling numbers of the second kind. Via Schur-Weyl duality, these dimensions equal the multiplicities of the irreducible $\mathsf{S}_n$-modules in $\mathsf{M}_n^{\otimes k}$. Our dimension expressions hold for any $n \geq 1$ and $k\ge0$. Our methods are based on an analog of Frobenius reciprocity that we show holds for the centralizer algebras of arbitrary finite groups and their subgroups acting on a finite-dimensional module. This enables us to generalize the above result to various analogs of the partition algebra including the centralizer algebra for the alternating group acting on $\mathsf{M}_n^{\otimes k}$ and the quasi-partition algebra corresponding to tensor powers of the reflection representation of $\mathsf{S}_n$., Comment: 27 page main document with 4 page appendix; Similar to version 1, but with new author, a new positive formula for Theorem 5.5, and a new bijective proof in Section 5.3

We consider the following counting problem related to the card game SET: How many $k$-element SET-free sets are there in an $n$-dimensional SET deck? Through a series of algebraic reformulations and reinterpretations, we show the answer to this question satisfies two polynomiality conditions., Comment: 8 pages, largely expository

We define a $q$-deformation of the classical ring of integer-valued polynomials which we call the ring of quantum integer-valued polynomials. We show that this ring has a remarkable combinatorial structure and enjoys many positivity properties: for instance, the structure constants for this ring with respect to its basis of $q$-binomial coefficient polynomials belong to $\mathbb{N}[q]$. We then classify all maps from this ring into a field, extending a known classification in the classical case where $q=1$., Comment: 32 pages, 1 figure

We give new interpretations of the Deligne categories $\underline{Rep}(GL_t)$ and $\underline{Rep}(S_t)$ (and their abelian envelopes) over $\mathbb{C}$ in terms of modular representations of general linear and symmetric groups of large rank in large characteristic. In particular we make sense of the sentence "$\underline{Rep}(S_n)$ is the limit of $Rep(S_{p+n})$ over $\bar{\mathbb{F}}_p$ as $p$ goes to infinity". We then give examples of how to pass results between these different settings., Comment: 10 pages

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 $\lambda$-rings and Brauer characters., Comment: 20 pages, latex; added Proposition 2.9, Remark 2.11 in the new version; subsection 3.4 rewritten, small error pointed out by K. Coulembier corrected in proof of Proposition 3.6 and a another small error in Remark 3.3

We study asymptotic properties of the modular representation theory of symmetric groups and investigate modular analogs of stabilization phenomena in characteristic zero. The main results are equivalences of categories between certain abelian subcategories of representations of $S_n$ and $S_m$ for different $n$ and $m$. We apply these results to obtain a structural result for $FI$-modules, and to prove a result conjectured by Deligne in a recent letter to Ostrik., Comment: Removed a unnecessary restriction on characteristic. Fixed some few minor errors and clarified the exposition in places

Working in the setting of Deligne categories, we generalize a result of Marin that hooks generate the representation ring of symmetric groups to wreath products of symmetric groups with a fixed finite group or Hopf algebra. In particular, when we take the finite group to be cyclic order 2 we recover a conjecture of Marin about Coxeter groups in type B., Comment: 11 pages

We consider the isoperimetric problem in planar sectors with density $r^{p}$, and with density $a>1$ inside the unit disk and $1$ outside. We characterize solutions as a function of sector angle. We also solve the isoperimetric problem in $\mathbb{R}^{n}$ with density $r^{p},\; p<0$., Comment: 29 pages

We provide very general symmetrization theorems in arbitrary dimension and codimension, in products, warped products, and certain fiber bundles such as lens spaces, including Steiner, Schwarz, and spherical symmetrization and admitting density., Comment: 9 pages