Recently Sarah Bockting-Conrad introduced the double lowering operator $ψ$ for a tridiagonal pair. Motivated by $ψ$ we consider the following problem about polynomials. Let $\mathbb F$ denote an algebraically closed field. Let $x$ denote an indeterminate, and let $\mathbb F\lbrack x \rbrack$ denote the algebra consisting of the polynomials in $x$ that have all coefficients in $\mathbb F$. Let $N$ denote a positive integer or $\infty$. Let $\lbrace a_i\rbrace_{i=0}^{N-1}$, $\lbrace b_i\rbrace_{i=0}^{N-1}$ denote scalars in $\mathbb F$ such that $\sum_{h=0}^{i-1} a_h \not= \sum_{h=0}^{i-1} b_h$ for $1 \leq i \leq N$. For $0 \leq i \leq N$ define polynomials $τ_i, η_i \in \mathbb F\lbrack x \rbrack$ by $τ_i = \prod_{h=0}^{i-1} (x-a_h)$ and $η_i = \prod_{h=0}^{i-1} (x-b_h)$. Let $V$ denote the subspace of $\mathbb F\lbrack x \rbrack$ spanned by $\lbrace x^i\rbrace_{i=0}^N$. An element $ψ\in \operatorname{End}(V)$ is called double lowering whenever $ψτ_i \in \mathbb F τ_{i-1}$ and $ψη_i \in \mathbb F η_{i-1}$ for $0 \leq i \leq N$, where $τ_{-1}=0$ and $η_{-1}=0$. We give necessary and sufficient conditions on $\lbrace a_i\rbrace_{i=0}^{N-1}$, $\lbrace b_i\rbrace_{i=0}^{N-1}$ for there to exist a nonzero double lowering map. There are four families of solutions, which we describe in detail.
The $X=M$ conjecture asserts that the $1D$ sum and the fermionic formula coincide up to some constant power. In the case of type $A,$ both the $1D$ sum and the fermionic formula are closely related to Kostka polynomials. Double Kostka polynomials $K_{\Bla,\Bmu}(t),$ indexed by two double partitions $\Bla,\Bmu,$ are polynomials in $t$ introduced as a generalization of Kostka polynomials. In the present paper, we consider $K_{\Bla,\Bmu}(t)$ in the special case where $\Bmu=(-,μ'').$ We formulate a $1D$ sum and a fermionic formula for $K_{\Bla,\Bmu}(t),$ as a generalization of the case of ordinary…
The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integral representation of the classical $_2F_1$ function. Based on a $q$-series transformation formula and the Nassrallah--Rahman integral we prove a $q$--beta integral which has twelve parameters, with several other results, both classical and new, included as special cases. This $q$-beta integral also allows us to derive a curious double $q$--series transformation formula, which includes one formula of Al--Salam and Ism…
Let $K_{(m,p)}$ denote the family of double twist knots where $2m-1$ and $2p$ are non-zero integers denoting the number of half-twists in each region. Using a result of Takata, we prove a formula for the colored Jones polynomial of $K_{(-m,-p)}$ and $K_{(-m,p)}$. The latter case leads to new families of $q$-hypergeometric series generalizing the Kontsevich-Zagier series. We also use Bailey pairs and formulas of Walsh to find cyclotomic-like expansions for the colored Jones polynomials of $K_{(m,p)}$ and $K_{(m,-p)}$.
We study the compatibility of the action of the DAHA of type GL with two inverse systems of polynomial rings obtained from the standard Laurent polynomial representations. In both cases, the crucial analysis is that of the compatibility of the action of the Cherednik operators. Each case leads to a representation of a limit structure (the +/- stable limit DAHA) on a space of almost symmetric polynomials in infinitely many variables (the standard representation). As an application, we show that the defining representation of the double Dyck path algebra arises from the standard representation o…
In the paper, we describe the Drinfel'd double structure of the $n$-rank Taft algebra and all of its simple modules, and then endow its $R$-matrices with some application to knot invariants. The knot invariants we get is a generalization of the Jones polynomial, in particular, it recovers the Jones polynomial in rank $1$ case, while in rank $2$ case, it is the one-parameter specialization of the two-parameter unframed Dubrovnik polynomial, and in higher rank case it is the composite ($n$-power) of the Jones polynomial.
In this paper, we propose and discuss implications of a general conjecture that there is a canonical action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K \subset S^3$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct 3-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev in \cite{RT90}. We also deduce …
We study a topological aspect of rank-1 double affine Hecke algebra (DAHA). Clarified is a relationship between the DAHA of A1-type (resp. CC1-type) and the skein algebra on a once-punctured torus (resp. a 4-punctured sphere), and the SL(2;Z) actions of DAHAs are identified with the Dehn twists on the surfaces. Combining these two types of DAHA, we construct the DAHA representation for the skein algebra on a genus-two surface, and we propose a DAHA polynomial for a double-torus knot, which is a simple closed curve on a genus two Heegaard surface in S3. Discussed is a relationship between the D…
