Double Hall-Littlewood symmetric polynomials
Jiayi Chen, Ming Lu, Shiquan Ruan
TL;DR
The paper constructs a deep link between the derived Hall algebra of the Jordan quiver and the ring of double symmetric functions ${\mathcal{D}}\Lambda_t$, via a robust raising/lowering operator framework that defines double Hall-Littlewood functions $V_{\lambda,\mu}$ parameterized by bipartitions. It proves an explicit isomorphism ${\Phi}_t$ sending natural basis elements to the double HL basis and derives horizontal and vertical Pieri rules for these functions, with specializations to HL ($t$ general) and to Schur-Laurent at $t=0$. The work further develops generating functions in the derived Hall algebra, establishes key identities connecting ${\widetilde{E}}(y,z)$, ${\widetilde{H}}(y,z)$, and ${\widetilde{\Theta}}(y,z)$, and transfers these relations to ${\mathcal{D}}\Lambda_t$, enabling transfer of transition relations between the algebraic and symmetric-function sides. Overall, the results extend Hall-Littlewood theory to a double-variable setting and provide tools for studying derived Hall algebras through double symmetric functions. The framework unifies categorical Hall structures with bi-graded symmetric function theory, offering new Pieri rules and generating-function identities with potential applications in representation theory and algebraic combinatorics.
Abstract
We establish a ring isomorphism between the derived Hall algebra of the Jordan quiver and the ring of double symmetric functions (i.e., the ring of symmetric polynomials in two sets of countably many variables, invariant under the respective actions of their symmetric groups) with a parameter $t$. This isomorphism maps the derived Hall basis (the natural basis of the derived Hall algebra) to a class of double Hall-Littlewood (HL) symmetric functions, which are formulated via raising and lowering operators. These double HL functions are parameterized by bipartitions; they reduce to the classical HL functions when one of the partitions is empty, and specialize to Schur Laurent symmetric functions at $t = 0$. We also derive the Pieri rules for these double HL functions. Additionally, we obtain several natural generating functions for the derived Hall algebra as well as their transition relations, which can be transferred to the ring of double symmetric functions via the established ring isomorphism.