A. P. Isaev, A. I. Molev
We show that all primitive idempotents for the Brauer algebra B_n(w) can be found by evaluating a rational function in several variables which has the form of a product of R-matrix type factors. This provides an analogue of the fusion procedure for B_n(w).
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We establish two versions of the fusion procedure for the walled Brauer algebras. In each of them, a complete system of primitive pairwise orthogonal idempotents for the walled Hecke algebra is constructed by consecutive evaluations of a rational function in several variables on contents of standard walled tableaux.
In this paper we prove an analogue of Brauer's theorem for faithful objects in fusion categories. Other notions, such as the order and the index associated to faithful objects of fusion categories are also discussed. We show that the index of a faithful simple object of a fusion categories coincides with the order of the universal grading group of the fusion category.
We give a new fusion procedure for the Brauer algebra by showing that all primitive idempotents can be found by evaluating a rational function in several variables which has the form of a product of R-matrix type factors. In particular, this provides a new fusion procedure for the symmetric group involving an arbitrary parameter. The R-matrices are solutions of the Yang--Baxter equation associated with the classical Lie algebras g_N of types B, C and D. Moreover, we construct an evaluation homomorphism from a reflection equation algebra B(g_N) to U(g_N) and show that the fusion procedure provi…
Let $p$ be a prime number, $G$ a finite group, $P$ a $p$-subgroup of $G$ and $k$ an algebraically closed field of characteristic $p$. We study the relationship between the category $\Ff_P(G)$ and the behavior of $p$-permutation $kG$-modules with vertex $P$ under the Brauer construction. We give a sufficient condition for $\Ff_P(G)$ to be a saturated fusion system. We prove that for Scott modules with abelian vertex, our condition is also necessary. In order to obtain our results, we prove a criterion for the categories arising from the data of $(b, G)$-Brauer pairs in the sense of Alperin-Brou…
Let $k$ be an algebraically closed field of prime characteristic $p$ and $P$ a finite $p$-group. We compute the Scott $kG$-module with vertex $P$ when $\mathcal{F}$ is a constrained fusion system on $P$ and $G$ is Park's group for $\mathcal{F}$. In the case $\mathcal{F}$ is a fusion system of the quadratic group $Qd(p)=(\mathbb{Z}/p \times \mathbb{Z}/p)\rtimes {\mathrm{SL}}(2,p)$ on a Sylow $p$-subgroup $P$ of $Qd(p)$ and $G$ is Park's group for $\mathcal{F}$, we prove that the Scott $kG$-module with vertex $P$ is Brauer indecomposable.
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。
Traceless projection of mixed tensor products, and walled Brauer algebras
本文提出了一种基于 Schur-Weyl 对偶构造混合张量积无迹投影的自包含方法，将其识别为中心化代数中的幂等元，并推广至半单墙型 Brauer 代数及 Hermite 空间应用。
We present an approach of calculating the group of braided autoequivalences of the category of representations of the Drinfeld double of a finite dimensional Hopf algebra $H$ and thus the Brauer-Picard group of $H$-$\mathrm{mod}$. We consider two natural subgroups and a subset as candidates for generators. In this article $H$ is the group algebra of a finite group $G$. As our main result we prove that any element of the Brauer-Picard group, fulfilling an additional cohomological condition, decomposes into an ordered product of our candidates. For elementary abelian groups $G$ our decomposition…
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…
A construction of bases for cell modules of the Birman--Murakami--Wenzl (or B--M--W) algebra $B_n(q,r)$ by lifting bases for cell modules of $B_{n-1}(q,r)$ is given. By iterating this procedure, we produce cellular bases for B--M--W algebras on which a large abelian subalgebra, generated by elements which generalise the Jucys--Murphy elements from the representation theory of the Iwahori--Hecke algebra of the symmetric group, acts triangularly. The triangular action of this abelian subalgebra is used to provide explicit criteria, in terms of the defining parameters $q$ and $r$, for B--M--W alg…
To any block idempotent $b$ of a group algebra $kG$ of a finite group $G$ over a field $k$ of characteristic $p>0$, Puig associated a fusion system and proved that it is saturated if the $k$-algebra $kC_G(P)e$ is split, where $(P,e)$ is a maximal $kGb$-Brauer pair. We investigate in the non-split case how far the fusion system is from being saturated by describing it in an explicit way as being generated by the fusion system of a related block idempotent over a larger field together with a single automorphism of the defect group.
A complete system of pairwise orthogonal minimal idempotents for Birman-Murakami-Wenzl algebras is obtained by a consecutive evaluation of a rational function in several variables on sequences of quantum contents of up-down tableaux. A by-product of the construction is a one-parameter family of fusion procedures for Hecke algebras. Classical limits to two different fusion procedures for Brauer algebras are described.
We use the fusion formulas of the symmetric group and of the Hecke algebra to construct solutions of the Yang-Baxter equation on irreducible representations of $\mathfrak{gl}_N$, $\mathfrak{gl}_{N|M}$, $U_q(\mathfrak{gl}_N)$ and $U_q(\mathfrak{gl}_{N|M})$. The solutions are obtained via the fusion procedure for the Yang--Baxter equation, which is reviewed in a general setting. Distinguished invariant subspaces on which the fused solutions act are also studied in the general setting, and expressed, in general, with the help of a fusion function. Only then, the general construction is specialise…
We derive a new expression for the diagonal matrix elements of irreducible representations of the symmetric group. We obtain this new expression using Cherednik's fusion procedure. However, instead of splitting Young diagrams into their rows or columns, we consider their principal hooks. This minimises the number of auxiliary parameters needed in the fusion procedure.
We derive a new expression for the q-analogue of the Young symmetrizer which generate irreducible representations of the Hecke algebra. We obtain this new expression using Cherednik's fusion procedure. However, instead of splitting Young diagrams into their rows or columns, we consider their principal hooks. This minimises the number of auxiliary parameters needed in the fusion procedure.
Finite dimensional irreducible modules of the two-parameter quantum enveloping algebra $U_{r,s}(\mathfrak{sl}_n)$ are explicitly constructed using the fusion procedure when $rs^{-1}$ is generic. This provides an alternative and combinatorial description of the Schur-Weyl duality for the two-parameter quantum linear algebras of type $A$.
A complete system of primitive pairwise orthogonal idempotents for the Coxeter groups of type $B$ and, more generally, for the complex reflection groups $G(m,1,n)$ is constructed by a sequence of evaluations of a rational function in several variables with values in the group ring. The evaluations correspond to the eigenvalues of the two arrays of Jucys--Murphy elements.
We give a new version of the fusion procedure for the symmetric group which originated in the work of Jucys and was developed by Cherednik. We derive it from the Jucys-Murphy formulas for the diagonal matrix units for the symmetric group.
The fusion procedure provides a way to construct new solutions to the Yang-Baxter equation. In the case of the symmetric group the fusion procedure has been used to construct diagonal matrix elements using a decomposition of the Young diagram into its rows or columns. We present a new construction which decomposes the diagram into hooks, the great advantage of this is that it minimises the number of auxiliary parameters needed in the procedure. We go on to use the hook fusion procedure to find diagonal matrix elements computationally and calculate supporting evidence to a previous conjecture. …
Hook fusion procedure for hyper-octahedral groups
本研究将Grime的对称群hook融合程序应用于超八面体群，推导得到其不可约表示对角矩阵元的新表达式，可减少融合过程所需辅助参数数量。
Hook fusion procedure for direct product of symmetric groups
本文利用对称群的 Grime 钩融合程序推导出直积群 $S_r\ imes S_s$ 不可约表示对角矩阵元的新表达式，通过减少辅助参数简化了融合过程，并扩展该方法以构造完备的正交本原幂等元集合。
The primitive idempotents of the generic degenerate cycloctomic Hecke algebras are derived by consecutive evaluations of a certain rational function. This rational function depends only on the Specht modules and the normalization factors are the weights of the Brundan-Kleshchev trace.
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.
Inspired by the work [IMOg2], in this note, we prove that the pairwise orthogonal primitive idempotents of generic cyclotomic Birman-Murakami-Wenzl algebras can be constructed by consecutive evaluations of a certain rational function. In the appendix, we prove a similar result for generic cyclotomic Nazarov-Wenzl algebras.
On the Jucys-Murphy method and fusion procedure for the Sergeev superalgebra
本文利用Jucys-Murphy元素为Sergeev超代数构造完备本原幂等元，给出单模半正规形式及基向量显式构造，并证明这些幂等元可由新版本融合程序得到。
A complete system of primitive pairwise orthogonal idempotents for cyclotomic Hecke algebras is constructed by consecutive evaluations of a rational function in several variables on quantum contents of multi-tableaux. This function is a product of two terms, one of which depends only on the shape of the multi-tableau and is proportional to the inverse of the corresponding Schur element.
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.
Fusion for the Yang-Baxter equation and the braid group
本文基于WinterBraids X课程讲义，阐述Yang-Baxter方程的融合过程，并由此构造出融合Hecke代数这一新代数例子。
We give explicit realizations of irreducible representations of the Yangian of the general linear Lie algebra and of its twisted analogues, corresponding to symplectic and orthogonal Lie algebras. In particular, we develop the fusion procedure for twisted Yangians. For the non-twisted Yangian, this procedure goes back to the works of Cherednik.
We give a new construction of primitive idempotents of the Hecke algebras associated with the symmetric groups. The idempotents are found as evaluated products of certain rational functions thus providing a new version of the fusion procedure for the Hecke algebras. We show that the normalization factors which occur in the procedure are related to the Ocneanu--Markov trace of the idempotents.
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