Representation Theory
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Showing new listings for Monday, 29 June 2026
- [1] arXiv:2606.27668 [pdf, other]
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Title: Representations of infinite speciesComments: 38 page, comments welcomeSubjects: Representation Theory (math.RT)
We consider species, consisting of a possibly infinite set of rings, and bimodules between them. Simson realised the category of representations as a functor category, which we prove is hereditary when each of the rings is semisimple. We use purity to provide sufficient conditions, in order for a representation to decompose into indecomposables with local endomorphism rings. For any bifunctor valued in bimodules, we functorially construct species equipped with commutativity conditions. This generates examples coming from a range of topics, such as subobject lattices in abelian length categories, the field choice problem in persistent homology, and topological field theories with defects.
- [2] arXiv:2606.27707 [pdf, html, other]
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Title: The representations of the Lie superalgebra $p(3)$ in characteristic 3Comments: arXiv admin note: substantial text overlap with arXiv:2601.15932Subjects: Representation Theory (math.RT)
Let $g$ be the Lie superalgebra $p(3)$ of rank 2 over an algebraically closed field $K$ of characteristic $p=3$. We classify all irreducible modules of $g$, and give the character formulae for irreducible modules.
- [3] arXiv:2606.27879 [pdf, html, other]
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Title: The brick chain complexity of an artin algebraComments: 9 pagesSubjects: Representation Theory (math.RT)
We consider the category of finitely generated modules over an artin algebra $A$. It is known that any module $M$ has a brick chain filtration. We say that M has brick chain complexity at most $t$ provided $M$ has a brick chain filtration of length at most $t$. The brick chain complexity of A is by definition the supremum of the brick chain complexity of the indecomposable $A$-modules. The aim of this note is to calculate the brick chain complexity for some algebras. We will exhibit algebras with arbitrarily large brick chain complexity.
- [4] arXiv:2606.28085 [pdf, html, other]
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Title: Average divisibility in character tables of $\mathrm{GL}_2(\mathbb{F}_q)$Comments: Version 1: 22 pagesSubjects: Representation Theory (math.RT); Combinatorics (math.CO); Number Theory (math.NT)
Let $q$ range over odd prime powers and let $G_q=\mathrm{GL}_2(\mathbb{F}_q)$. Fix a prime number $\ell$. Motivated by work of Peluse and Soundararajan on Miller's conjecture for character tables of symmetric groups, we study the proportion of entries in the character table of $G_q$ which are not divisible by $\ell$, in the sense of divisibility in the ring of algebraic integers. We prove that $N_\ell(q)=\frac{q^4}{2}+O_\epsilon(q^{3+\epsilon})$ for every $\epsilon>0$, where $N_\ell(q)$ denotes the number of entries which are not divisible by $\ell$. We also show that the number of zero entries is $\frac{q^4}{2}+O_\epsilon(q^{3+\epsilon})$. Consequently, the proportion of all entries not divisible by $\ell$ tends to $1/2$, while the proportion of nonzero entries not divisible by $\ell$ tends to $1$. This differs significantly from the symmetric-group case, where almost every character-table entry is divisible by any fixed prime. We also prove an angular equidistribution result for the nonzero character values as $q\to\infty$. We show that the arguments become equidistributed in $[0,2\pi]$. This proves an analogue of Miller's question on the distribution of signs among the nonzero entries in character tables of symmetric groups.
- [5] arXiv:2606.28091 [pdf, html, other]
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Title: Stability of the exterior cube $γ$-factors for $\mathrm{GL}(6)$Subjects: Representation Theory (math.RT); Number Theory (math.NT)
We prove the stability of the Langlands-Shahidi local $\gamma$-factor for the exterior cube representation of $\mathrm{GL}_6$. More precisely, if $\pi_1$ and $\pi_2$ are irreducible admissible generic representations of $\mathrm{GL}_6(F)$ with the same central character, then \[ \gamma(s,\pi_1\otimes\chi,\wedge^3,\psi)= \gamma(s,\pi_2\otimes\chi,\wedge^3,\psi) \] for every sufficiently ramified character $\chi$ of $F^\times$, where $\chi$ is regarded as a character of $\mathrm{GL}_6(F)$ through the determinant. The proof uses the realization of the exterior cube representation by the maximal parabolic subgroup of the simply connected group of type $E_6$. We give an explicit description of the relevant geometric quotient $U_M\backslash N'$, compute its invariant measure, and relate Shahidi's partial Bessel functions to partial Bessel integrals on the Levi subgroup. The desired stability then follows from an asymptotic expansion of these partial Bessel integrals and the vanishing of highly ramified Mellin transforms.
- [6] arXiv:2606.28096 [pdf, other]
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Title: On $U(\mathfrak{h})$-free modules of finite rank over $\mathfrak{sl}(2)$Comments: 41 pagesSubjects: Representation Theory (math.RT)
We study $\mathfrak{sl}(2)$-modules that are free of finite rank over $U(\mathfrak h)$, where $\mathfrak h$ is a fixed Cartan subalgebra of $\mathfrak{sl}(2)$. These modules form a natural class of non-weight modules. The coherent families obtained from this class via the weighting functor are identified. We also study a distinguished class of indecomposable $U(\mathfrak h)$-free modules defined in terms of Jordan blocks and give a recursive description of their socle filtrations. Finally, we apply the general results to exponential modules arising from the first Weyl algebra and obtain simplicity criteria for these modules.
New submissions (showing 6 of 6 entries)
- [7] arXiv:2606.27497 (cross-list from math.CO) [pdf, html, other]
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Title: Enumerating matrices with prescribed entries in an adjoint orbitComments: 29 pagesSubjects: Combinatorics (math.CO); Algebraic Geometry (math.AG); Representation Theory (math.RT)
We study intersections of conjugacy classes of square matrices over a finite field with affine coordinate subspaces, or equivalently matrices in a fixed adjoint orbit with prescribed entries. Our main result treats the case of prescribed columns: for a partially defined linear map we give a Hall scalar product formula for the number of extensions to an endomorphism with prescribed similarity invariants. This formula is expressed in terms of skew modified Hall--Littlewood functions and $q$-Whittaker functions. As applications, we count monic matrix polynomials over $\mathbb{F}_q$ with prescribed Smith normal form and with prescribed determinant, and recover the Gerstenhaber--Reiner formula for the number of square matrices with a fixed characteristic polynomial. We also note that known point-count formulas for Hessenberg varieties imply related formulas for Hessenberg supports involving chromatic quasisymmetric functions, motivating polynomiality questions for more general supports and prescribed affine slices.
- [8] arXiv:2606.27523 (cross-list from math.CO) [pdf, html, other]
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Title: Coordinate projections of $c$-vectors of cluster algebras from the annulusComments: 17 pages, 1 figure. Ancillary files: exact-integer Python verification code reproducing the computational classificationSubjects: Combinatorics (math.CO); Rings and Algebras (math.RA); Representation Theory (math.RT)
For an acyclic cluster algebra, the $c$-vectors are, up to sign, the real
Schur roots of the associated root system. We study the two-coordinate
projections $(c_v, c_w)$ of this configuration: when the difference
$c_v - c_w$ is bounded the image lies in a band of lattice lines, and we ask
when the projection fills that band. A band-existence dichotomy, valid in
every acyclic type, shows the difference is bounded if and only if the null
root satisfies $\delta_v = \delta_w$. For affine type $\widetilde{A}_n$ (the
annulus), in the source-sink orientation, we resolve the filling question
completely: every coordinate projection fills its band except along the
source-sink diagonal, which carries only the finite regular part. The
obstruction is the Auslander--Reiten defect, which a projection sees on its
diagonal exactly when the defect is a coordinate difference; the only such
pair is the source-sink pair of $\widetilde{A}_n$, so the pattern depends on
the chosen seed. More generally, every banded pair of null-root coefficient
one fills, except these diagonals. Off the diagonal a banded pair in
$\widetilde{E}_7$ fails to fill, so non-filling is not confined to type
$\widetilde{A}_n$; a computation classifies the pairs of coefficient at least
two over a range of affine types, where this $\widetilde{E}_7$ pair is the
only further failure, and the general classification remains open. - [9] arXiv:2606.27749 (cross-list from math.NT) [pdf, html, other]
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Title: On Franke's theorem in the simplest caseComments: 11 pages. Comments are welcomeSubjects: Number Theory (math.NT); Representation Theory (math.RT); Spectral Theory (math.SP)
For level one spherical automorphic forms on the upper half-plane, we prove directly that every automorphic form is a sum of a cusp form and a linear combination of Laurent coefficients of the standard Eisenstein series. This is the simplest instance of Franke's general theorem, which asserts that automorphic forms on a reductive group are spanned by Laurent coefficients of Eisenstein series induced from cuspidal automorphic forms on Levi subgroups. Unlike Franke's general argument, ours does not invoke Langlands' construction of the discrete automorphic spectrum from cuspidal Eisenstein series. It rests instead on basic analytic properties of automorphic forms and Green's identity.
- [10] arXiv:2606.27901 (cross-list from math.QA) [pdf, html, other]
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Title: Zonal Spherical Functions of Quantum Symmetric PairsSubjects: Quantum Algebra (math.QA); Representation Theory (math.RT)
We identify the zonal and character spherical functions for quantum symmetric pairs with the symmetric Koornwinder--Macdonald polynomials. To this end, the methods of Letzter's 2004 paper are translated to modern conventions and right coideal subalgebras, and extended to non-standard cases and cases with non-reduced restricted root systems. For the last elusive Satake type, FII, a conjecture is provided.
- [11] arXiv:2606.28140 (cross-list from math-ph) [pdf, html, other]
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Title: Shirokov realizations of low dimensional Lie algebrasComments: 28 pagesSubjects: Mathematical Physics (math-ph); Representation Theory (math.RT)
We compute the transitive realizations for the low dimensional cases of real Lie algebras up to dimension four using Shirokov's method. First, the generic realizations are given, then, making use of the known list of subalgebras, nongeneric realizations are computed. The result is compared with the known classification of Popovych et al.
Cross submissions (showing 5 of 5 entries)
- [12] arXiv:2407.10119 (replaced) [pdf, other]
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Title: Affine and cyclotomic Schur categoriesComments: v3:minor corrections, final published version, to appear in Transformation GroupsSubjects: Representation Theory (math.RT); Quantum Algebra (math.QA)
Using the affine web category introduced in a prequel as a building block, we formulate a diagrammatic $\Bbbk$-linear monoidal category, the affine Schur category, for any commutative ring $\Bbbk$. We then formulate diagrammatic categories, the cyclotomic Schur categories, with arbitrary parameters at positive integral levels. Integral bases consisting of elementary diagrams are obtained for affine and cyclotomic Schur categories. A second diagrammatic basis, called a double SST basis, for any such cyclotomic Schur category is also established, leading to a conjectural higher level RSK correspondence. We show that the endomorphism algebras with the double SST bases are isomorphic to degenerate cyclotomic Schur algebras with their cellular bases, providing a first diagrammatic presentation of the latter. The presentations for the affine and cyclotomic Schur categories are much simplified when $\Bbbk$ is a field of characteristic zero.
- [13] arXiv:2411.18427 (replaced) [pdf, other]
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Title: Brick chain filtrationsComments: Assertion 6.1 has been corrected. There are several minor improvements. A (new) example of an algebra with infinite brick chain complexity will be contained in a separate noteSubjects: Representation Theory (math.RT)
We deal with the category of finitely generated modules over an artin algebra $A$. Recall that an object in an abelian category is said to be a brick provided its endomorphism ring is a division ring. Simple modules are, of course, bricks, but in case $A$ is connected and not local, there do exist bricks which are not simple. The aim of this survey is to focus the attention to filtrations of modules where all factors are bricks, with bricks being ordered in some definite way.
In general, a module category will have many oriented cycles. Recently, Demonet has proposed to look at so-called brick chains in order to deal with a very interesting directedness feature of a module category. These are the orderings of bricks which we will use.
This is a survey which relies on recent investigations by a quite large group of mathematicians. We have singled out some important observations and have reordered them in order to obtain a completely self-contained (and elementary) treatment of the relevance of bricks in a module category. (Most of the papers we rely on are devoted to what is called $\tau$-tilting theory, but for the results we are interested in, there is no need to deal with $\tau$-tilting, or even with the Auslander-Reiten translation $\tau$). - [14] arXiv:2511.08779 (replaced) [pdf, other]
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Title: Morita equivalences between cyclotomic KLR algebras in types $\mathtt{C}_\infty$ and $\mathtt{A}_\infty$Comments: 18 pages. v2 has several improvements after comments from a refereeSubjects: Representation Theory (math.RT)
We prove that level one cyclotomic KLR algebras in type $\mathtt{C}_\infty$ are graded Morita equivalent to level two cyclotomic KLR algebras in type $\mathtt{A}_\infty$. We hence deduce the graded decomposition numbers and full submodule structures of all level one cyclotomic KLR algebras in type $\mathtt{C}_\infty$.
- [15] arXiv:2511.18218 (replaced) [pdf, html, other]
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Title: Classification of simple commutative algebras in the Delannoy categoryComments: 24 pagesSubjects: Representation Theory (math.RT)
The Delannoy category is an interesting pre-Tannakian category associated to the oligomorphic group $\mathbb{G}$ of automorphisms of the totally ordered set $(\mathbf{R}, <)$. By construction, it admits some obvious simple commutative algebras, corresponding to certain transitive $\mathbb{G}$-sets. We show that these account for all of the simple commutative algebras in the Delannoy category. Previous results of this kind have been limited to interpolation categories; since the Delannoy category cannot be obtained by interpolation, new methods are required.
- [16] arXiv:2604.00290 (replaced) [pdf, html, other]
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Title: The Drinfeld center of an oligomorphic tensor categoryComments: 54 pagesSubjects: Representation Theory (math.RT)
Recently, Harman and the second author introduced a new construction of pre-Tannakian tensor categories based on oligomorphic groups. We develop tools for analyzing the Drinfeld centers of these categories, and compute the center explicitly in a number of cases. In particular, we find several finitely tensor-generated pre-Tannakian categories (including the Delannoy category) that are identified with their own center via the canonical functor; prior to this work, we knew no such examples besides the category of vector spaces.
- [17] arXiv:2311.16327 (replaced) [pdf, html, other]
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Title: A Hodge filtration on chiral homology and Poisson homology of associated schemesComments: 27 pages; published in Forum MathematicumSubjects: Quantum Algebra (math.QA); Algebraic Geometry (math.AG); Representation Theory (math.RT)
We introduce filtrations in chiral homology complexes of smooth elliptic curves, exploiting the mixed Hodge structure on cohomology groups of configuration spaces. We use these to relate the chiral homology of a smooth elliptic curve with coefficients in a vertex algebra with the Poisson homology of the associated Poisson scheme. As an application we deduce finite dimensionality results for chiral homology in low degrees.
- [18] arXiv:2511.17034 (replaced) [pdf, html, other]
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Title: Affine Jacobi-Trudi Identities and $q,t$-Rogers-Ramanujan IdentitiesJournal-ref: SIGMA 22 (2026), 062, 50 pagesSubjects: Combinatorics (math.CO); Number Theory (math.NT); Representation Theory (math.RT)
We conjecture affine or Hall-Littlewood analogues of the dual Jacobi-Trudi identities for orthogonal and symplectic Schur functions indexed by rectangular partitions of maximal height. These conjectures are then used to derive $t$-analogues of many known Rogers-Ramanujan identities for the characters of standard modules of affine Lie algebras. This includes $t$-analogues of the classical Rogers-Ramanujan identities, (some of) the Andrews-Gordon identities and the $\mathrm{C}_n^{(1)}$, $\mathrm{A}_{2n}^{(2)}$ and $\mathrm{D}_{n+2}^{(2)}$ GOW identities. We also prove an affine analogue of the dual Jacobi-Trudi identity for Schur functions indexed by rectangular partitions of arbitrary height.
- [19] arXiv:2602.19209 (replaced) [pdf, html, other]
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Title: SemiringsComments: 34 pages with a few minor corrections: includes 61-entry bibliographySubjects: Rings and Algebras (math.RA); Representation Theory (math.RT)
We survey theory developed over the past 10 years of semirings which need not be additively cancellative. The main features are a specified ``null ideal'' $\mcA_0$ of a semiring $\mcA,$ taking the place of a zero element, and a ``surpassing relation,'' taking the place of equality, which permit generalizations of the classical algebraic theory to polynomials and their roots, algebraic geometry, matrices, linear algebra, varieties, categories, and module theory. The ``pair'' $(\mcA,\mcA_0)$ is studied along the lines of universal algebra.
- [20] arXiv:2605.28770 (replaced) [pdf, html, other]
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Title: Cutoff profiles for conjugacy invariant random walks on symmetric groupsComments: v2: 27 pages, 5 figures, comments welcome!Subjects: Probability (math.PR); Combinatorics (math.CO); Representation Theory (math.RT)
We prove asymptotic equivalents of characters for finite-level representations of symmetric groups, that is, for Young diagrams which have all but finitely many boxes on their first row. The proofs rely on computing the number of ribbon tableaux of different types, which allows us to estimate characters via the Murnaghan--Nakayama rule. We deduce that random walks on symmetric groups generated by conjugacy classes with a macroscopic number of fixed points have a Poissonian cutoff profile. We also prove that the random involution walk exhibits cutoff and find its cutoff profile. Finally, we obtain numerics for the random transposition walk on a deck of 52 cards, giving concrete estimates on the question that originally motivated Diaconis and Shahshahani.
- [21] arXiv:2606.26776 (replaced) [pdf, other]
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Title: Artin monoids, their homomorphisms and twinsComments: AmsLaTeX+amsrefs, 79 pages; some misprints corrected. Parts of this paper appeared previously in the preprint arXiv:2405.18821. The current paper has been substantially reorganized and includes new approachesSubjects: Quantum Algebra (math.QA); Combinatorics (math.CO); Representation Theory (math.RT)
Motivated by the twin homomorphism problem for Coxeter groups and the corresponding Hecke monoids, we find a large class of its solutions originating from standard homomorphisms of Artin monoids and their compositions. These homomorphisms are expected to be injective when they are optimal and injective on generators, which generalizes the homogeneous homomorphisms and the famous Tits conjecture settled by Crisp and Paris. We classify disjoint standard homomorphisms and conjecture the complete classification when the domain is of rank two.