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Register a new userOn the freeness of the cyclotomic BMW algebras: admissibility and an isomorphism with the cyclotomic Kauffman tangle algebras
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The cyclotomic Birman-Murakami-Wenzl (BMW) algebras B_n^k, introduced by R. Haring-Oldenburg, are a generalisation of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (aka Ariki-Koike algebras) and type B knot theory. In this paper, we prove the algebra is free and of rank k^n (2n-1)!! over ground rings with parameters satisfying so-called admissibility conditions. These conditions are necessary in order for these results to hold and originally arise from the representation theory of B_2^k, which is analysed by the authors in a previous paper. Furthermore, we obtain a geometric realisation of B_n^k as a cyclotomic version of the Kauffman tangle algebra, in terms of affine n-tangles in the solid torus, and produce explicit bases that may be described both algebraically and diagrammatically. The admissibility conditions are the most general offered in the literature for which these results hold; they are necessary and sufficient for all results for general n.
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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.
----- Please see the pdf file for the actual abstract and important remarks, which could not be put here due to the arXiv length restrictions. ----- This thesis presents a study of the cyclotomic BMW (Birman-Murakami-Wenzl) algebras, introduced by Haring-Oldenburg as a generalization of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. They are shown to be free of rank k^n (2n-1)!! and to have a topological realization as a certain cylindrical analogue of the Kauffman Tangle algebra. Furthermore, the cyclotomic BMW algebras are proven to be cellular, in the sense of Graham and Lehrer. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007.
Inspired by the work [PA], we establish an explicit algebra isomorphism between the degenerate cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}(q)$ and a direct sum of matrix algebras over tensor products of degenerate cyclotomic Hecke algebras of type $A$. We then develop several applications of this result, including a new proof of the modular representation theory of $Y_{r,n}^{d}(q)$, a semisimplicity criterion for it and cellularity of it. Moreover, we prove that $Y_{r,n}^{d}(q)$ is a symmetric algebra and determine the associated Schur elements by using the isomorphism theorem for it.
Let $Sc(vL)$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $He_{n,r}$, introduced by Dipper-James-Mathas. In this paper, we consider $v$-decomposition numbers of $Sc(vL)$, namely decomposition numbers with respect to the Jantzen filtrations of Weyl modules. We prove, as a $v$-analogue of the result obtained by Shoji-Wada, a product formula for $v$-decomposition numbers of $Sc(vL)$, which asserts that certain $v$-decomposition numbers are expressed as a product of $v$-decomposition numbers for various cyclotomic $q$-Schur algebras associated to Ariki-koike algebras $He_{n_i,r_i}$ of smaller rank. Moreover we prove a similar formula for $v$-decomposition numbers of $He_{n,r}$ by using a Schur functor.
Let $S$ be the cyclotomic $q$-Schur algebra associated to the Ariki-Koike algebra $H_{n,r}$ of rank $n$, introduced by Dipper-James-Mathas. For each $p = (r_1, ..., r_g)$ such that $r_1 + ... + r_g = r$, we define a subalgebra $S^p$ of $S$ and its quotient algebra $bar S^p$. It is shown that $S^p$ is a standardly based algebra and $bar S^p$ is a cellular algebra. By making use of these algebras, we show that certain decomposition numbers for $S$ can be expressed as a product of decomposition numbers for cyclotomic $q$-Schur algebras associated to smaller Ariki_koike algebras $H_{n_k,r_k}$.
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