Cyclotomic Nazarov-Wenzl Algebras

Abstract

Abstract. Nazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain "cyclotomic quotients" of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank r n (2n − 1)!! (when Ω is u-admissible). We next show that these algebras are cellular and give a labelling for the simple modules of the cyclotomic Nazarov-Wenzl algebras over an arbitrary field. In particular, this gives a construction of all of the finite dimensional irreducible modules of the affine Wenzl algebra.

Seminormal forms and Gram determinants for cellular algebras

“…♦ It should be possible to find JM-elements for other cellular algebras such as the partition algebras and the cyclotomic Nazarov-Wenzl algebras [2].…”

Degenerate two-boundary centralizer algebras

“…This work may proceed in a number of directions. First, an analogous theory may also be developed for centralizers of type B, C, and D, which will parallel that of the degenerate affine Wenzl algebra as studied in [Nazarov 1996;Ariki et al 2006]. Also, functorial techniques developed in [Orellana and Ram 2007] may be used to promote the study of calibrated Ᏼ ext k -modules, given in Section 5, to that of all standard modules.…”

Cyclotomic Carter-Payne homomorphisms

“…The general case is similar except that it requires more notation. Let ν = λ ∪ μ = λ [1] λ [2] ∪ μ [1] μ [2] set γ = |ν\λ| = γ 1 + γ 2 , where γ 1 = |λ [1] \μ [1] | and γ 2 = |λ [2] \μ [2] |. As in Example 3.38, define t ν λ to be the standard ν-tableau such that (t ν λ ) ↓n = t λ , the numbers n + 1, .…”

“…For each multipartition λ there is a graded Specht module S λ which is a Z-free R Λ n -module [6]. If K is a field then S λ ⊗ Z K is isomorphic to the graded Specht module constructed in [15,20] and, in turn, this module is a graded lift of the (ungraded) Specht module S λ of the cyclotomic Hecke algebras H Λ n [2,10]. In Definition 3.26 we give a purely combinatorial, but quite technical, condition for when a pair (λ, μ) of multipartitions is a cyclotomic Carter-Payne pair.…”

“…For example, in §3. 2 we introduce the notion of stubborn strings which we think will have wider application. Using these ideas we prove our Main Theorem as Theorem 3.28.…”