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Sebastien Palcoux

Publications and source records attributed to Sebastien Palcoux.

At least 19 recordsLinked to original sources
arXiv2026

Sharp bounds for intervals in finite subgroup lattices

Let H be a subgroup of a finite group G, and put n = [G:H] > 1. If p is the least prime divisor of n, we prove that the number of subgroups K with H <= K <= G is less than c(p) n^((log_p n)/4). Here c(p) is the product of (1 - p^(-j))^(-1) over all positive integers j, multiplied by the sum of p^(-z^2) over all integers z. The exponent and the constant are simultaneously optimal, already for elementary abelian p-groups of even rank. In particular, for arbitrary n > 1, the number of intermediate subgroups is less than 7.371968802 n^((log_2 n)/4). This removes the polynomial factor from the best previously known relative bound and extends the sharp absolute subgroup-counting theorem to arbitrary subgroup intervals. The group-theoretic argument is elementary. Its main ingredient is a weighted packing inequality for relative generating tuples, obtained from an escape tree on the coset space G/H.

math.GR↗
arXiv2026

Local Ocneanu rigidity for separable algebra objects

We establish local Ocneanu rigidity for separable algebra objects in Hom-finite monoidal categories C over an algebraically closed field k. A separability morphism contracts the first two Hochschild cohomology groups without requiring an abelian ambient category. Separable algebra structures and homomorphisms from separable sources have open algebraic-group orbits, and affine Bezout estimates give effective finiteness results. For Frobenius subalgebras, exchange relations replace the source and embedding data by a single self-dual idempotent. We prove that their separable inner-conjugacy classes are open in the exchange locus, yielding a bound of 2^(dim_k End_C(X)) for nonzero ambient algebras; connectedness gives the same bound on the actual number of subalgebras. As an application, we obtain an effective form of the Etingof-Walton finiteness theorem: every finite-dimensional semisimple Hopf algebra H over the complex numbers has at most 2^(dim_C H) left coideal subalgebras. In the unitary setting, we bound E-compatible intermediates of C*-algebra and von Neumann algebra inclusions up to unitary conjugacy, under finite-index and finite-center hypotheses. For irreducible subfactors, these improve the 9^[M:N] bound of Bakshi-Das-Liu-Ren to 2^[M:N].

math.QA↗
arXiv2025

Exchange relations and Frobenius subalgebras

Bisch and Jones established a bijection between the intermediate subfactors of an irreducible subfactor and certain idempotents satisfying exchange relations. In this paper, we generalize this result to Karoubian monoidal categories through Frobenius subalgebras. As an application, we show that certain morphisms on a C*-correspondence arise from intermediate C*-subalgebras if and only if they are averaging operators.

math.QA↗
arXiv2025

A new criterion for integral modular categorification

A generalization of an argument due to Etingof-Nikshych-Ostrik yields a highly efficient necessary criterion for integral modular categorification. This criterion allows us to complete the classification of categorifiable integral modular data up to rank 14, and up to rank 25 in the odd-dimensional case.

math.QA↗
arXiv2025

Classifying integral Grothendieck rings up to rank 5 and beyond

In this paper, we define a Grothendieck ring as a fusion ring categorifiable into a fusion category over the complex field. An integral fusion ring is called Drinfeld if all its formal codegrees are integers dividing the global Frobenius--Perron dimension. Every integral Grothendieck ring is necessarily Drinfeld. Using the fact that the formal codegrees of integral Drinfeld rings form an Egyptian fraction summing to 1, we derive a finite list of possible global FPdims for small ranks. Applying Normaliz, we classify all fusion rings with these candidate FPdims, retaining only those admitting a Drinfeld structure. To exclude Drinfeld rings that are not Grothendieck rings, we analyze induction matrices to the Drinfeld center, classified via our new Normaliz feature. Further exclusions and constructions involve group-theoretical fusion categories and Schur multipliers. Our main result is a complete classification of integral Grothendieck rings up to rank 5, extended to rank 7 in odd-dimensional and noncommutative cases using Frobenius--Schur indicators and Galois theory. Moreover, we show that any noncommutative, odd-dimensional, integral Grothendieck ring of rank at most 22 is pointed of rank 21. We also classify all integral 1-Frobenius Drinfeld rings of rank 6, identify the first known non-Isaacs integral fusion category (which turns out to be group-theoretical), classify integral noncommutative Drinfeld rings of rank 8, and integral 1-Frobenius MNSD Drinfeld rings of rank 9. Finally, we determine the smallest-rank exotic simple integral fusion rings: rank 4 in general, rank 6 in the Drinfeld case, and rank 7 in the 1-Frobenius Drinfeld case.

math.QA↗
arXiv2025

Frobenius subalgebra lattices in tensor categories

This paper generalizes Watatani's finiteness theorem for intermediate subfactors to a wide class of monoidal categories. We characterize the sublattices of Frobenius subalgebra posets in abelian monoidal categories by introducing a notion of ambient selfduality. By extending several key results---such as the planar algebraic exchange relation and Landau's theorems---to linear monoidal categories, we establish a structural rigidity property for a formal angle associated to every coherent pair of Frobenius subalgebras (that is, whose intersection and sum are ambiently selfdual). Furthermore, within a weak positivity framework, we deduce that such coherent sublattices are finite for any connected Frobenius algebra. This significantly generalizes Watatani's theorem, since the unitary Frobenius subalgebra lattices are inherently coherent through a property termed rigidity invariance. Applications of this work include a unified framework that encompasses several previously unrelated finiteness results. Specifically, we recover the finiteness of the left coideal subalgebra lattice of a finite-dimensional semisimple Hopf algebra (Etingof-Walton theorem) under a well-supported coherence hypothesis, as well as the finiteness of the intermediate C*-algebra lattice for a finite-index unital irreducible inclusion of C*-algebras (relaxing simplicity in Ino-Watatani theorem) under the E-compatibility condition shown to be unavoidable. Furthermore, we present a variety of new applications involving abstract spin chains and vertex operator algebras, alongside speculations on quantum arithmetic that include extensions of Ore's theorem, Euler's totient and sigma functions, and RH.

math.QA↗
arXiv2023

Burnside type results for fusion categories

In this paper, we extend a classical vanishing result of Burnside from the character tables of finite groups to the character tables of commutative fusion rings, or more generally to a certain class of abelian normalizable hypergroups. We also treat the dual vanishing result. We show that any nilpotent unitary fusion categories satisfy both Burnside's property and its dual. Using Drinfeld's map, we obtain that the Grothendieck ring of any weakly-integral modular fusion category satisfies both properties. As applications, we prove new identities that hold in the Grothendieck ring of any weakly-integral fusion category satisfying the dual-Burnside's property, thus providing new categorification criteria. In particular we improve [OY23, Theorem 4.5] as follows: A weakly integral modular fusion category of FPdim md with d square-free coprime with m and FPdim(X)^2 for every simple object X, has a pointed modular fusion subcategory of FPdim d. We also present new results on perfect modular fusion categories, including a Cauchy-type theorem.

math.QA↗
arXiv2023

Classification of integral modular data up to rank 13

This paper classifies the modular data of integral modular fusion categories up to rank 13, and integral half-Frobenius fusion rings up to rank 12. We establish that every perfect case within these bounds is trivial. Furthermore, we refine the non-pointed odd-dimensional modular data at ranks below 25 to exactly three items, all of rank 17, FPdim 225, and type [[1,3],[3,8],[5,6]], filling existing literature gaps. For rank 25, we narrow the perfect case to three open types. Our core insight is that Egyptian fractions, typically used to list possible types, can be chosen with squared denominators. We develop several type criteria as initial filters. To construct the fusion rings, we solve dimension and associativity equations utilizing custom-built features in Normaliz. S-matrices are generated by self-transposing the character table, and T-matrices are derived by solving the Anderson-Moore-Vafa equations, concluding with the verification of extended modular data axioms. From rank 13 onward, types are restricted by modular-specific properties involving universal grading, congruence representations of the modular group, and Galois action. This establishes critical arithmetic constraints: up to rank 21, a prime divisor of the global FPdim cannot exceed the rank, and up to rank 15 (non-pointed case), it cannot exceed half the rank. Ultimately, we reduce the rank 14 classification to 35 possible types, 8 of which are non-perfect.

math.QA↗
arXiv2022

Structure constants, Isaacs property and Extended Haagerup fusion categories

This paper presents an abstract Isaacs property that involves the Fourier transform for fusion rings, which may be non-commutative, thus expanding upon the commutative version described in [12]. A categorical version of this property was subsequently introduced in [8] for any spherical fusion category, matching with our abstract version in the pseudo-unitary case. We demonstrate that the Isaacs property occupies a distinct position, falling between the integrality of structure constants and the 1-Frobenius properties, in the commutative case. We show that the Extended Haagerup fusion categories, denoted as EHi, do not satisfy the Isaacs property. This finding provides a negative response to [8, Question 5.8], refutes [12, Conjecture 2.5], and recovers that EH1 lacks a braiding structure.

math.QA↗
arXiv2022

Complete Positivity of Comultiplication and Primary Criteria for Unitary Categorification

In this paper, we investigate quantum Fourier analysis on subfactors and unitary fusion categories. We prove the complete positivity of the comultiplication for subfactors and derive a primary $n$-criterion of unitary categorifcation of multifusion rings. It is stronger than the Schur product criterion when $n\geq3$. The primary criterion could be transformed into various criteria which are easier to check in practice even for noncommutative, high-rank, high-multiplicity, multifusion rings. More importantly, the primary criterion could be localized on a sparse set, so that it works for multifusion rings with sparse known data. We give numerous examples to illustrate the efficiency and the power of these criteria.

math.OA↗
arXiv2022

Triangular Prism Equations and Categorification

We introduce the triangular prism equations (TPE) for fusion categories, obtained by evaluating triangular prisms in terms of tetrahedra. Using an oriented graphical calculus, we show that the geometric symmetries of the regular tetrahedron are preserved. In the spherical case, we prove that the TPE are equivalent to the pentagon equations after a suitable change of basis. These equations provide new insight for managing complexity via localization. As a consequence, and using the Fuchs-Runkel-Schweigert theorem on the second Frobenius-Schur indicator, we obtain new categorification criteria. As an application, we solve all remaining open cases to complete the classification of unitary 1-Frobenius simple integral fusion categories up to rank 8 and up to Frobenius-Perron dimension 20000.

math.QA↗
arXiv2021

Interpolated family of non group-like simple integral fusion rings of Lie type

This paper is motivated by the quest of a non-group irreducible finite index depth 2 maximal subfactor. We compute the generic fusion rules of the Grothendieck ring of Rep(PSL(2,q)), q prime-power, by applying a Verlinde-like formula on the generic character table. We then prove that this family of fusion rings R_q interpolates to all integers q>=2, providing (when q is not prime-power) the first example of infinite family of non group-like simple integral fusion rings. Furthermore, they pass all the known criteria of (unitary) categorification. This provides infinitely many serious candidates for solving the famous open problem of whether there exists an integral fusion category which is not weakly group-theoretical. We prove that a complex categorification (if any) of an interpolated fusion ring R_q (with q non prime-power) cannot be braided, and so its Drinfeld center must be simple. In general, this paper proves that a non-pointed simple fusion category is non-braided if and only if its Drinfeld center is simple; and also that every simple integral fusion category is weakly group-theoretical if and only if every simple integral modular fusion category is pointed.

math.QA↗
arXiv2020

Classification of Grothendieck rings of complex fusion categories of multiplicity one up to rank six

This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six. Among 72 possible fusion rings, $25$ ones are filtered out by using categorification criteria. Each of the remaining 47 fusion rings admits a unitary complex categorification. We found 6 new Grothendieck rings, categorified by applying a localization approach of the Pentagon Equation.

math.CT↗
arXiv2019

Boolean lattices in finite alternating and symmetric groups

Given a group $G$ and a subgroup $H$, we let $\mathcal{O}_G(H)$ denote the lattice of subgroups of $G$ containing $H$. This paper provides a classification of the subgroups $H$ of $G$ such that $\mathcal{O}_{G}(H)$ is Boolean of rank at least $3$, when $G$ is a finite alternating or symmetric group. Besides some sporadic examples and some twisted versions, there are two different types of such lattices. One type arises by taking stabilizers of chains of regular partitions, and the other type arises by taking stabilizers of chains of regular product structures. As an application, we prove in this case a conjecture on Boolean overgroup lattices, related to the dual Ore's theorem and to a problem of Kenneth Brown.

math.GR↗
arXiv2019

Fusion Bialgebras and Fourier Analysis

We introduce fusion bialgebras and their duals and systematically study their Fourier analysis. As an application, we discover new efficient analytic obstructions on the unitary categorification of fusion rings. We prove the Hausdorff-Young inequality, uncertainty principles for fusion bialgebras and their duals. We show that the Schur product property, Young's inequality and the sum-set estimate hold for fusion bialgebras, but not always on their duals. If the fusion ring is the Grothendieck ring of a unitary fusion category, then these inequalities hold on the duals. Therefore, these inequalities are analytic obstructions of categorification. We classify simple integral fusion rings of Frobenius type up to rank 8 and of Frobenius-Perron dimension less than 4080. We find 34 ones, 4 of which are group-like and 28 of which can be eliminated by applying the Schur product property on the dual. In general, these inequalities are obstructions to subfactorize fusion bialgebras.

math.QA↗
arXiv2017

Ore's theorem on cyclic subfactor planar algebras and beyond

Ore proved that a finite group is cyclic if and only if its subgroup lattice is distributive. Now, since every subgroup of a cyclic group is normal, we call a subfactor planar algebra cyclic if all its biprojections are normal and form a distributive lattice. The main result generalizes one side of Ore's theorem and shows that a cyclic subfactor is singly generated in the sense that there is a minimal 2-box projection generating the identity biprojection. We conjecture that this result holds without assuming the biprojections to be normal, and we show that it is true for small lattices. We finally exhibit a dual version of another theorem of Ore and a non-trivial upper bound for the minimal number of irreducible components for a faithful complex representation of a finite group.

math.OA↗