Fused permutations algebras and degenerate affine Hecke algebras

This paper gives an algebraic presentation of an algebra called the fused permutations algebra in the one-boundary case. It is obtained through a detailed study of the degenerate cyclotomic Hecke algebra. In particular, we prove that the fused permutations algebra is a quotient of the degenerate cyclotomic affine Hecke algebra, and we also describe a basis combinatorially in terms of signed permutations with avoiding patterns. In order to understand this quotient, we study the primitive idempotents of this degenerate cyclotomic affine Hecke algebra.

💡 Research Summary

The paper provides a comprehensive algebraic description of the fused permutations algebra Hₖ,ₙ in the one‑boundary case, establishing it as a concrete quotient of the degenerate cyclotomic affine Hecke algebra ˆH(κ₁,κ₂)ₙ. After recalling the classical Schur–Weyl duality and the role of the symmetric group algebra C Sₙ as the N‑independent centraliser of GLₙ on V^{⊗n}, the author introduces Hₖ,ₙ as the centraliser of GLₙ acting on Sᵏ(V)⊗V^{⊗n}. Two equivalent definitions of Hₖ,ₙ are given: a diagrammatic one based on “fused permutations” (graphs with k parallel edges attached to the first point) and an algebraic one as the double‑sandwich Pₖ C S_{k+n} Pₖ, where Pₖ is the symmetriser idempotent of C Sₖ. An explicit isomorphism H: Pₖ ω Pₖ ↦