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Register a new userA combinatorial formula expressing periodic $R$-polynomials
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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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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.
We present a decomposition of the generalized binomial coefficients associated with Jack polynomials into two factors: a stem, which is described explicitly in terms of hooks of the indexing partitions, and a leaf, which inherits various recurrence properties from the binomial coefficients and depends exclusively on the skew diagram. We then derive a direct combinatorial formula for the leaf in the special case where the two indexing partitions differ by at most two rows. This formula also exhibits an unexpected symmetry with respect to the lengths of the two rows.
Our investigation in the present paper is based on three important results. (1) In [12], Ringel introduced Hall algebra for representations of a quiver over finite fields and proved the elements corresponding to simple representations satisfy the quantum Serre relation. This gives a realization of the nilpotent part of quantum group if the quiver is of finite type. (2) In [4], Green found a homological formula for the representation category of the quiver and equipped Ringels Hall algebra with a comultiplication. The generic form of the composition subalgebra of Hall algebra generated by simple representations realizes the nilpotent part of quantum group of any type. (3) In [9], Lusztig defined induction and restriction functors for the perverse sheaves on the variety of representations of the quiver which occur in the direct images of constant sheaves on flag varieties, and he found a formula between his induction and restriction functors which gives the comultiplication as algebra homomorphism for quantum group. In the present paper, we prove the formula holds for all semisimple complexes with Weil structure. This establishes the categorification of Greens formula.
In this paper, we generalize the categorifical construction of a quantum group and its canonical basis introduced by Lusztig (cite{Lusztig,Lusztig2}) to the generic form of the whole Ringel-Hall algebra. We clarify the explicit relation between the Green formula in cite{Green} and the restriction functor in cite{Lusztig2}. By a geometric way to prove the Green formula, we show that the Hopf structure of a Ringel-Hall algebra can be categorified under Lusztigs framework.
We use the isomorphisms between the $R$-matrix and Drinfeld presentations of the quantum affine algebras in types $B$, $C$ and $D$ produced in our previous work to describe finite-dimensional irreducible representations in the $R$-matrix realization. We also review the isomorphisms for the Yangians of these types and use Gauss decomposition to establish an equivalence of the descriptions of the representations in the $R$-matrix and Drinfeld presentations of the Yangians.
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