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We show that the $imath$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 $imath$Hall algebra to the ring of symmetric functions in two parameters $t, theta$, which maps the $imath$Hall basis to a class of (modified) inhomogeneous Hall-Littlewood ($imath$HL) functions. The (modified) $imath$HL functions admit a formulation via raising and lowering operators. We formulate and prove Pieri rules for (modified) $imath$HL functions. The modified $imath$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.
For any triple $(i,a,mu)$ consisting of a vertex $i$ in a quiver $Q$, a positive integer $a$, and a dominant $GL_a$-weight $mu$, we define a quiver current $H^{(i,a)}_mu$ acting on the tensor power $Lambda^Q$ of symmetric functions over the vertices
We establish automorphisms with closed formulas on quasi-split $imath$quantum groups of symmetric Kac-Moody type associated to restricted Weyl groups. The proofs are carried out in the framework of $imath$Hall algebras and reflection functors, thanks
We show that the generic Hall algebra of nilpotent representations of an oriented cycle specialised at $q=0$ is isomorphic to the generic extension monoid in the sense of Reineke. This continues the work of Reineke.
Expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical (i.e., quasi-parabolic KL) bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possib
$imath$quantum groups are generalizations of quantum groups which appear as coideal subalgebras of quantum groups in the theory of quantum symmetric pairs. In this paper, we define the notion of classical weight modules over an $imath$quantum group,