P. Bantay
We present an overview of the close analogies between the character rings of finite groups and the fusion rings of rational conformal models, which follow from general principles related to orbifold deconstruction.
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We compute the fusion rings of positive energy representations of the loop groups of the simple, simply connected Lie groups.
We study the fusion rings of tilting modules for a quantum group at a root of unity modulo the tensor ideal of negligible tilting modules. We identify them in type A with the combinatorial rings from [KS] and give a similar description of the sp(2n)-fusion ring in terms of noncommutative symmetric functions. Moreover we give a presentation of all fusion rings in classical types as quotients of polynomial rings extending known results in special cases. Finally we also compute the fusion rings for type G2.
We study fusion rings for degenerate minimal models ($p=q$ case) for N=0 and N=1 (super)conformal algebras. We consider a distinguished family of modules at the level $c=1$ and $c=3/2$ and show that the corresponding fusion rings are isomorphic to the representation rings for ${\mathfrak sl}(2, {\bf C})$ and ${\mathfrak osp}(1|2)$ respectively.
For any normal commutative Hopf subalgebra $K=k^G$ of a semisimple Hopf algebra we describe the ring inside $kG$ obtained by the restriction of $H$-modules. If $G=\Z_p$ this ring determines a fusion ring and we give a complete description for it. The case $G=\Z_{p^n}$ and some other applications are presented.
Coxeter planes as fixed points of Verlinde fusion rings
本文通过超群作用构造ADE类型Coxeter群的Coxeter平面作为固定点，基于融合环的正整数模分类。
In this note we describe a general elementary procedure to attach a fusion ring to any Kac-Moody algebra of affine type. In the case of untwisted affine algebras, they are usual fusion rings in the literature. In the case of twisted affine algebras, they are exactly the twisted fusion rings defined by the author in [Ho2] via tracing out diagram automorphisms on conformal blocks for appropriate simply-laced Lie algebras. We also relate the fusion ring to the modular S-matrix for any Kac-Moody algebra of affine type.
From a certain induced representation $\mathcal{P}_\ell$ of a double affine Weyl group, we construct a ring $\mathcal{F}_\ell$ that is isomorphic to the fusion ring, or Verlinde algebra, associated to affine Lie algebras at fixed positive integer level for both twisted and untwisted type. The induced representation, which also has a natural commutative associative algebra structure and is modular invariant with respect to certain congruence subgroups, contains $\mathcal{F}_\ell$ as an ideal and we show how it naturally inherits the modular invariance property from $\mathcal{P}_\ell.$ This cons…
The integral group rings $\mathbb{Z}G$ for finite groups $G$ are precisely those fusion rings whose basis elements have Frobenius-Perron dimension 1, and each is categorifiable in the sense that it arises as the Grothendieck ring of a fusion category. Here we analyze the structure and representation theory of fusion rings with a basis of elements whose Frobenius-Perron dimensions take exactly one value distinct from 1. Our goal is a set of results which assist in characterizing when such fusion rings are categorifiable. As proof of concept, we completely classify the categorifiable near-group …
Let p be a prime, M be a finite group, F be the field with p elements, and V be an absolutely irreducible FM-module. Then V has a universal deformation ring R(M,V) whose structure is closely related to the first and second cohomology groups of M with coefficients in Hom_F(V,V). We consider the case when M is an extension of a dihedral group G whose order is relatively prime to p by an elementary abelian p-group N of rank 2. We determine the cohomology groups and also R(M,V) for various V and show to what extent R(M,V) sees the fusion of N in M.
Representations of Coxeter groups over fusion rings and hyperplane complements
本文研究Coxeter群在融合环上的忠实实现及其Vinberg系统，证明其诱导超平面补集嵌入，从而为Artin–Tits群间的强可容许同态提供几何实现。
We study the fusion semirings arising from easy quantum groups. We classify all the possible free ones, answering a question of T. Banica and R. Vergnioux : these are exactly the fusion rings of quantum groups without any nontrivial one-dimensional representation. We then classify the possible groups of one-dimensional representations for free easy quantum groups. As an application, we give a unified proof of the Haagerup property for a broad class of easy quantum groups, recovering as special cases previous results by M. Brannan and F. Lemeux. We end with some considerations on the descriptio…
A simple, combinatorial construction of the sl(n)-WZNW fusion ring, also known as Verlinde algebra, is given. As a byproduct of the construction one obtains an isomorphism between the fusion ring and a particular quotient of the small quantum cohomology ring of the Grassmannian Gr(k,k+n). We explain how our approach naturally fits into known combinatorial descriptions of the quantum cohomology ring, by establishing what one could call a `Boson-Fermion-correspondence' between the two rings. We also present new recursion formulae for the structure constants of both rings, the fusion coefficients…
For a finite group $G$, we define a ghost ring and a mark homomorphism for the double Burnside ring of left-free $(G,G)$-bisets. In analogy to the case of the Burnside ring $B(G)$, the ghost ring has a much simpler ring structure, and after tensoring with $\QQ$ one obtains an isomorphism of $\QQ$-algebras. As an application of a key lemma, we obtain a very general formula for the Brauer construction applied to a tensor product of two $p$-permutation bimodules $M$ and $N$ in terms of Brauer constructions of the bimodules $M$ and $N$. Over a field of characteristic 0 we determine the simple modu…
We construct a well-behaved transfer map from the p-local Burnside ring of the underlying p-group S to the p-local Burnside ring of a saturated fusion system F. Using this transfer map, we give new results on the characteristic idempotent of F -- the unique idempotent in the p-local double Burnside ring of S satisfying properties of Linckelmann and Webb. We describe this idempotent explicitly both in terms of fixed points and as a linear combination of transitive bisets. Additionally, using fixed points we determine the map for Burnside rings given by multiplication with the characteristic ide…
This is a proceedings article reviewing a recent combinatorial construction of the su(n) WZNW fusion ring by C. Stroppel and the author. It contains one novel aspect: the explicit derivation of an algorithm for the computation of fusion coefficients different from the Kac-Walton formula. The discussion is presented from the point of view of a vertex model in statistical mechanics whose partition function generates the fusion coefficients. The statistical model can be shown to be integrable by linking its transfer matrix to a particular solution of the Yang-Baxter equation. This transfer matrix…
We consider the Burnside ring $A(\mathcal{F})$ of $\mathcal{F}$-stable $S$-sets for a saturated fusion system $\mathcal{F}$ defined on a $p$-group $S$. It is shown by S. P. Reeh that the monoid of $\mathcal{F}$-stable sets is a free commutative monoid with canonical basis $\{α_P\}$. We give an explicit formula that describes $α_P$ as an $S$-set. In the formula we use a combinatorial concept called broken chains which we introduce to understand inverses of modified Möbius functions.
Fusion Rings over Drinfeld Doubles
本文研究了Drinfeld双代数上的融合环，证明了二面体群和四面体群的乘积自由性，并揭示了其与共形场论的关联。
Rotation fixed points of undecorated braid varieties and fusion rings of affine Lie algebras
本文证明无装饰编织簇上由循环旋转诱导的 Zamolodchikov 变换的不动点栈的粗模空间同构于相应仿射李代数的融合环的谱，并推广至扭曲 Coxeter 元素，从而得到所有仿射李代数的融合环。
We prove that there is a finite level-independent bound on the number of relations defining the fusion ring of positive energy representations of the loop group of a simple, simply connected Lie group. As an illustration, we compute the fusion ring of $G_2$ at all levels.
Interpolated family of non group-like simple integral fusion rings of Lie type
本文利用类 Verlinde 公式计算 Rep(PSL(2,q)) 的通用融合规则并构造插值族 R_q，得到非群状单整融合环无限族，揭示其范畴化、Drinfeld 中心与弱群论性质的等价判据。
Classifying Grothendieck rings of pivotal fusion categories of rank four
本文对复数域上秩四枢轴融合范畴的Grothendieck环进行了完整分类，确定恰好十五个且均可酉范畴化，并利用新障碍与穷举方法给出维数上界3600。
Representation Rings of Fusion Systems and Brauer Characters
通过模特征理论技术研究融合系统的复特征环，证明Semeraro猜想对非奇异融合系统成立，并给出分解矩阵和Cartan矩阵的类比。
We describe a purely group-theoretic condition on an element g of a finite group G which implies that g has coefficient zero in every central idempotent element of the group ring RG, provided that R is a ring of prime characteristic. We use this condition to prove that the fusion system associated to a block of an alternating group is always isomorphic to the group fusion system of an alternating group.
Classifying integral Grothendieck rings up to rank 5 and beyond
本文定义了可范畴化的整Grothendieck环，利用Normaliz工具、Frobenius-Schur指标和Galois理论，完成了秩至5及部分高秩情况的完整分类，并发现了首个非Isaacs整fusion category等新结构。
Affine Pieri rule for periodic Macdonald spherical functions and fusion rings
本研究针对非扭及扭仿射李代数（不含$\u200d\widehat{BC}_n$型），提出周期Macdonald球面函数基的仿射Pieri法则，该法则在$\u200d\hat{A}_{n-1}$型下可重现已知结果并退化为共形场论融合环的经典Pieri公式。
Burnside type results for fusion categories
本文研究了融合范畴中Burnside型结果的扩展，证明了nilpotent单位融合范畴满足Burnside性质及其对偶，并提供了新的范畴化标准和应用。
The $\mathbb{Z}/2\mathbb{Z}$--graded intertwining operators are introduced. We study these operators in the case of ``degenerate'' N=1 minimal models, with the central charge $c=3/2$. The corresponding fusion ring is isomorphic to the Grothendieck ring for the Lie superalgebra ${\mathfrak osp}(1|2)$. We also discus multiplicity--2 fusion rules and logarithmic intertwiners.
This paper explains the relation between the fusion product of symmetric power sl(n) evaluation modules, as defined by Feigin and Loktev, and the graded coordinate ring R(mu), which describes the cohomology ring of the flag variety Fl(mu) of GL(N). The graded multiplicity spaces appearing in the decomposition of the fusion product into irreducible sl(n)-modules are identified with the multiplicity spaces of the Specht modules in R(mu). This proves that the Kostka polynomial gives the character of the fusion product in this case. In the case of the product of fundamental evaluation modules, we …
Categorification of based modules over the complex representation ring of $S_4$
该研究对对称群$S_4$复表示环上秩≤5的不可约有基模分类，得到16个不等价模，并基于此探讨了由子群射影表示诱导的模范畴实现的有基模范畴化问题。
The fusion-stable tom Dieck homomorphism
本文证明了tom Dieck同态及其因子符号同态在限制到p-群的融合稳定子群时仍满射，进而解决了Mazza-Miller的问题，确定了Lefschetz同态满射当且仅当域为F₂。
We return to the fusion rules for the Drinfeld double of the duals of the generalised Taft algebras that we studied in [9]. We first correct some proofs and statements in [9] that were incorrect, using stable homomorphisms. We then complete this with new results on fusion rules for the modules we had not studied in [9] and a classification of endotrivial and algebraic modules.
Motivic factorisation of KZ local systems and deformations of representation and fusion rings
本文证明KZ局部系统在点聚合时的动机分解，由此构造表示环与融合环的形变族，并给出$\mathfrak{sl}_n$情形计算算法，证明特定条件下整体单值有限且标量。
In this paper we expand on previous results, studying the extent to which one can detect fusion in certain finite groups $Γ$, from information about the universal deformation rings of absolutely irreducible $\mathbb{F}_pΓ$-modules. We consider groups which are extensions of finite irreducible subgroups of $Gl_2({\mathbb{C}})$ by elementary abelian $p$-groups of rank $2$.
This paper is the third of a series giving a self-contained way from the Neveu-Schwarz algebra to a new series of irreducible subfactors. Here we introduce the local von Neumann algebra of the Neveu-Schwarz algebra, to obtain Jones-Wassermann subfactors for each representation of the discrete series. Then using primary fields we prove the irreducibility of these subfactors; to next compute the Connes fusion ring and obtain the explicit formula of the subfactors indices.
Let G be a finite group. A complete system of pairwise orthogonal idempotents is constructed for the wreath product of G by the symmetric group by means of a fusion procedure, that is by consecutive evaluations of a rational function with values in the group ring. This complete system of idempotents is indexed by standard Young multi-tableaux. Associated to the wreath product of G by the symmetric group, a Baxterized form for the Artin generators of the symmetric group is defined and appears in the rational function used in the fusion procedure.
Reconstructing algebra objects from NIM-reps in pointed, near-group and quantum group-like fusion categories
本文利用 NIM-reps 研究三类融合范畴中代数的 Morita 等价类，并与现有分类结果对比，附录检验了秩≤4的模张量范畴的指数猜想。
We establish a correspondence between the levels of Verlinde rings and the depths of subfactors. Given the $l$-level Verlinde ring $R_l(G)$ of a simple compact Lie group $G$, the tensor products of fundamental representations give us the inclusion of a pair of $\text{II}_1$ factors $N\subset M$. For the depth $d$ of $N\subset M$, we first prove $d=l$ for type $A_n,C_n$ and $B_2$. More generally, the depth $d$ is shown to satisfy $β\cdot l\leq d\leq l$ with $β\in (0,1)$, where $β$ is uniquely determined by the simple type of $G$. We also show that the simple $N$-$N$-bimodules contained in $L^2(…
We study character rings of quasireductive Lie superalgebras and give a new proof of the Sergeev-Veselov theorem describing the character rings of finite-dimensional Kac-Moody superalgebras.
Characters and fusion rules of boundary W-algebras
本文研究例外W-代数的q-特征标与模数据，建立边界W-代数间的相等性，从而确定A型例外W-代数的融合规则。
Continuing our previous work, we determine a minimal left characteristic biset $X$ for every exotic fusion system $\mathcal{F}$ on the extraspecial group $S$ of order $7^3$ and exponent $7$ discovered by Ruiz and Viruel, and analyze the finite group $G$ obtained from $X$ which realizes $\mathcal{F}$ as the full subcategory of the $7$-fusion system of $G$. In particular, we obtain an upper bound for the exoticity index for $\mathcal{F}$.
Fusion invariant characters of p-groups
本文将Bárcenas--Cantarero定理推广到非饱和融合系，证明p-群的不可分解F-不变特征标数不小于F-共轭类数，并构造反例否定Cantarero--Combariza猜想。
In this paper we show that integral fusion categories with rational structure constants admit a natural group of symmetries given by the Galois group of their character tables. We also generalize a well known result of Burnside from representation theory of finite groups. More precisely, we show that any row corresponding to a non invertible object in the character table of a weakly integral fusion category contains a zero entry.
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