融合辫子与$U_q(gl_N)$张量表示的中心化子
Fused braids and centralisers of tensor representations of $U_q(gl_N)$
AI总结：
本文提出融合置换代数及其形变融合Hecke代数，证明$U_q(gl_N)$自然表示对称幂张量积的Schur–Weyl对偶，并刻画其不可约表示、分支规则及中心化子。
摘要：
本文提出了融合置换代数及其形变——融合Hecke代数。前者定义在一类我们称之为融合置换的组合对象上，其形变定义在一类我们称之为融合辫子的拓扑对象上。我们利用这些代数证明了$U_q(gl_N)$自然表示的任意对称幂的任意张量积的Schur–Weyl对偶定理。随后我们研究了融合Hecke代数，特别地，我们明确描述了其不可约表示和分支规则。最后，我们旨在对所考虑的$U_q(gl_N)$表示张量积的中心化子给出代数描述。我们构造了一个简单的显式元素，并猜想它生成从融合Hecke代数到中心化子的核。我们在若干情形下证明了这一猜想，特别地，我们得到了$U_q(sl_2)$任意有限维表示的任意张量积的中心化子的描述。
英文摘要：
We present in this paper the algebra of fused permutations and its deformation the fused Hecke algebra. The first one is defined on a set of combinatorial objects that we call fused permutations, and its deformation is defined on a set of topological objects that we call fused braids. We use these algebras to prove a Schur--Weyl duality theorem for any tensor products of any symmetrised powers of the natural representation of $U_q(gl_N)$. Then we proceed to the study of the fused Hecke algebras and in particular, we describe explicitely the irreducible representations and the branching rules. Finally, we aim to an algebraic description of the centralisers of the tensor products of $U_q(gl_N)$-representations under consideration. We exhibit a simple explicit element that we conjecture to generate the kernel from the fused Hecke algebra to the centraliser. We prove this conjecture in some cases and in particular, we obtain a description of the centraliser of any tensor products of any finite-dimensional representations of $U_q(sl_2)$.
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