Zhao and Xu (2013) constructed a functor from <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathfrak{o}(n)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">o</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>-Mod to <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathfrak{o}(n+2)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">o</mi> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>-Mod. In this paper, we use the functor successively to obtain full conformal oscillator representation of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathfrak{o}(2n+2)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">o</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> in n(n + 1) variables and determine the corresponding finite-dimensional irreducible module explicitly when the highest weight is dominant integral. We also find an equation of counting the dimension of an irreducible <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathfrak{o}(2n+2)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">o</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>-module in terms of certain alternating sum of the dimensions of irreducible <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathfrak{o}(2n)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="fraktur">o</mi> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>n</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>-modules, which leads to new combinatorial identities of classical type in the case of the Steinberg modules. One can use the results to study tensor decomposition of finite-dimensional irreducible modules by solving certain first-order linear partial differential equations, and thereby obtain the corresponding physically interested Clebsch–Gordan coefficients and exact solutions of Knizhnik–Zamolodchikov equation in WZW model of conformal field theory.

Full Conformal Oscillator Representations of $\mathfrak{o}(2n+2)$ and Combinatorial Identities

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发表日期：

2026年1月15日

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Zhenyu Zhou

摘要

Zhao and Xu (2013) constructed a functor from $\mathfrak{o}(n)$ $o (n)$ -Mod to $\mathfrak{o}(n+2)$ $o (n + 2)$ -Mod. In this paper, we use the functor successively to obtain full conformal oscillator representation of $\mathfrak{o}(2n+2)$ $o (2 n + 2)$ in n(n + 1) variables and determine the corresponding finite-dimensional irreducible module explicitly when the highest weight is dominant integral. We also find an equation of counting the dimension of an irreducible $\mathfrak{o}(2n+2)$ $o (2 n + 2)$ -module in terms of certain alternating sum of the dimensions of irreducible $\mathfrak{o}(2n)$ $o (2 n)$ -modules, which leads to new combinatorial identities of classical type in the case of the Steinberg modules. One can use the results to study tensor decomposition of finite-dimensional irreducible modules by solving certain first-order linear partial differential equations, and thereby obtain the corresponding physically interested Clebsch–Gordan coefficients and exact solutions of Knizhnik–Zamolodchikov equation in WZW model of conformal field theory.

Full Conformal Oscillator Representations of \(\mathfrak{o}(2n+2)\) and Combinatorial Identities

摘要