<p>We show that the <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\imath $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ı</mi> </math></EquationSource> </InlineEquation>Hall algebra of the Jordan quiver is a polynomial ring in infinitely many generators and obtain transition relations among several generating sets. We establish a ring isomorphism from this <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\imath $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ı</mi> </math></EquationSource> </InlineEquation>Hall algebra to the ring of symmetric functions in two parameters <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$t, \theta $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo>,</mo> <mi>θ</mi> </mrow> </math></EquationSource> </InlineEquation>, which maps the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\imath $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ı</mi> </math></EquationSource> </InlineEquation>Hall basis to a class of (modified) inhomogeneous Hall-Littlewood (<InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\imath $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ı</mi> </math></EquationSource> </InlineEquation>HL) functions. The (modified) <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\imath $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ı</mi> </math></EquationSource> </InlineEquation>HL functions admit a formulation via raising and lowering operators. We formulate and prove Pieri rules for (modified) <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\imath $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ı</mi> </math></EquationSource> </InlineEquation>HL functions. The modified <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\imath $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ı</mi> </math></EquationSource> </InlineEquation>HL functions specialize at <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\theta =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> to the modified HL functions; they specialize at <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\theta =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> to the deformed universal characters of type C, which further specialize at (t=0, <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\theta =1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>θ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) to the universal characters of type C.</p>

$\imath $Hall algebra of Jordan quiver and $\imath $Hall-Littlewood functions

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发表日期：

2025年11月6日

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Ming Lu,
Shiquan Ruan,
Weiqiang Wang

摘要

We show that the $\imath $ $ı$ 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 $\imath $ $ı$ Hall algebra to the ring of symmetric functions in two parameters $t, \theta $ $t, θ$ , which maps the $\imath $ $ı$ Hall basis to a class of (modified) inhomogeneous Hall-Littlewood ( $\imath $ $ı$ HL) functions. The (modified) $\imath $ $ı$ HL functions admit a formulation via raising and lowering operators. We formulate and prove Pieri rules for (modified) $\imath $ $ı$ HL functions. The modified $\imath $ $ı$ HL functions specialize at $\theta =0$ $θ = 0$ to the modified HL functions; they specialize at $\theta =1$ $θ = 1$ to the deformed universal characters of type C, which further specialize at (t=0, $\theta =1$ $θ = 1$ ) to the universal characters of type C.

\(\imath \)Hall algebra of Jordan quiver and \(\imath \)Hall-Littlewood functions

摘要