A. I. Molev
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.
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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.
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. …
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…
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。
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).
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。
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.
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.
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.
Remarks on symmetric fusion categories of low rank in positive characteristic
本文给出正特征下对称融合范畴秩的下界，证明第二 Adams 运算非恒等且满足特定条件的范畴为超 Tannakian，并分类了秩 3 及部分秩 4 的范畴。
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 $A$.
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.
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 study properties of symmetric fusion categories in characteristic $p$. In particular, we introduce the notion of a super Frobenius-Perron dimension of an object $X$ of such a category, and derive an explicit formula for the Verlinde fiber functor $F(X)$ of $X$ (defined by the second author) in terms of the usual and super Frobenius-Perron dimension of $X$. We also compute the decomposition of symmetric powers of objects of the Verlinde category, generalizing a classical formula of Cayley and Sylvester for invariants of binary forms. Finally, we show that the Verlinde fiber functor is unique…
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.
Classification of symmetric fusion categories over $\mathbb{R}$
该研究分类了实数域上的对称融合范畴，通过伽罗瓦下降将其约化为复域上对称融合范畴的半线性$\text{Z}_2$作用问题，建立了与带$\text{Z}_2 \times \text{BZ}_2$作用的有限群胚的塔纳卡-克赖因对应，得到了德利涅分类结果的实数类似物。
In this paper, we conjecture an extension of the Hilbert basis theorem and the finite generation of invariants to commutative algebras in symmetric finite tensor categories over fields of positive characteristic. We prove the conjecture in the case of semisimple categories and more generally in the case of categories with fiber functors to the characteristic $p > 0$ Verlinde category of $SL_{2}$. We also construct a symmetric finite tensor category $\sVec_{2}$ over fields of characteristic $2$ and show that it is a candidate for the category of supervector spaces in this characteristic. We fur…
Fusion-invariant representations for symmetric groups
本文研究对称群不变表示的分解唯一性问题，证明了对 $\mathbb{Z}/p \wr \mathbb{Z}/p$ 的唯一性当且仅当 $p=2$ 成立，并完全确定了 $p=2,3$ 时的表示环。
For a prime $p$, we describe a protocol for handling a specific type of fusion system on a $p$-group by computer. These fusion systems contain all saturated fusion systems. This framework allows us to computationally determine whether or not two subgroups are conjugate in the fusion system for example. We describe a generation procedure for automizers of every subgroup of the $p$-group. This allows a computational check of saturation. These procedures have been implemented using MAGMA. We describe a program to search for saturated fusion systems $\mathcal{F}$ on $p$-groups with $O_p(\mathcal{F…
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.
Orthogonal Howe duality and dynamical (split) symmetric pairs
受李代数相关动态结构启发，为分裂对称对引入动态融合、K-矩阵及Weyl群版本，研究(𝔰𝔬₂ₙ,Oₘ)-对偶性并证明𝔰𝔬₂ₙ侧标准算子与Oₘ侧对称对类似物可交换。
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.
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。
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