$ı$Hall algebra of Jordan quiver and $ı$Hall-Littlewood functions

Abstract

We show that the $ı$Hall algebra of the Jordan quiver is a polynomial ring in infinitely many generators and obtain transition relations among several generating sets. We establish a ring isomorphism from this $ı$Hall algebra to the ring of symmetric functions in two parameters $t,\theta$, which maps the $ı$Hall basis to a class of (modified) inhomogeneous Hall-Littlewood ($ı$HL) functions. The (modified) $ı$HL functions admit a formulation via raising and lowering operators. We formulate and prove Pieri rules for (modified) $ı$HL functions. The modified $ı$HL functions specialize at $\theta =0$ to the modified HL functions; they specialize at $\theta =1$ to the deformed universal characters of type C, which further specialize at $(t=0,\theta =1)$ to the universal characters of type C.

v2,41 pages,references added

References in corpus (3)

• $ı$Hall algebra of the projective line and $q$-Onsager algebra
• Deformed universal characters for classical and affine algebras
• Lectures on dualities ABC in representation theory

Cited by in corpus (1)

• Quantum Borcherds-Bozec algebras via semi-derived Ringel-Hall algebras