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Authors
Fabian Lenzen
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No Arabic abstract
Based on the NilHecke algebra $mathsf{NH}_n$, the odd NilHecke algebra developed by Ellis, Khovanov and Lauda and Kang, Kashiwara and Tsuchiokas quiver Hecke superalgebra, we develop the Clifford Hecke superalgebra $mathsf{NH}mathfrak{C}_n$ as another super-algebraic analogue of $mathsf{NH}_n$. We show that there is a notion of symmetric polynomials fitting in this picture, and we prove that these are generated by an appropriate analogue of elementary symmetric polynomials, whose properties we shall discuss in this text.
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In this paper, we study in the context of quantum vertex algebras a certain Clifford-like algebra introduced by Jing and Nie. We establish bases of PBW type and classify its $mathbb N$-graded irreducible modules by using a notion of Verma module. On the other hand, we introduce a new algebra, a twin of the original algebra. Using this new algebra we construct a quantum vertex algebra and we associate $mathbb N$-graded modules for Jing-Nies Clifford-like algebra with $phi$-coordinated modules for the quantum vertex algebra. We also show that the adjoint module for the quantum vertex algebra is irreducible.
Associated to the classical Weyl groups, we introduce the notion of degenerate spin affine Hecke algebras and affine Hecke-Clifford algebras. For these algebras, we establish the PBW properties, formulate the intertwiners, and describe the centers. We further develop connections of these algebras with the usual degenerate (i.e. graded) affine Hecke algebras of Lusztig by introducing a notion of degenerate covering affine Hecke algebras.
We introduce an odd double affine Hecke algebra (DaHa) generated by a classical Weyl group W and two skew-polynomial subalgebras of anticommuting generators. This algebra is shown to be Morita equivalent to another new DaHa which are generated by W and two polynomial-Clifford subalgebras. There is yet a third algebra containing a spin Weyl group algebra which is Morita (super)equivalent to the above two algebras. We establish the PBW properties and construct Verma-type representations via Dunkl operators for these algebras.
The notion of rational spin double affine Hecke algebras (sDaHa) and rational double affine Hecke-Clifford algebras (DaHCa) associated to classical Weyl groups are introduced. The basic properties of these algebras such as the PBW basis and Dunkl operator representations are established. An algebra isomorphism relating the rational DaHCa to the rational sDaHa is obtained. We further develop a link between the usual rational Cherednik algebra and the rational sDaHa by introducing a notion of rational covering double affine Hecke algebras.
In 1980, Lusztig introduced the periodic Kazhdan-Lusztig polynomials, which are conjectured to have important information about the characters of irreducible modules of a reductive group over a field of positive characteristic, and also about those of an affine Kac-Moody algebra at the critical level. The periodic Kazhdan-Lusztig polynomials can be computed by using another family of polynomials, called the periodic $R$-polynomials. In this paper, we prove a (closed) combinatorial formula expressing periodic $R$-polynomials in terms of the doubled Bruhat graph associated to a finite Weyl group and a finite root system.
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