AI 中文总结
该研究通过引入特定向量族，以新方式推导B、C、D型对偶Littlewood恒等式，获自由费米子构造证明相关测度是行列式形式，还建立广义Littlewood恒等式，包含著名恒等式特殊情况并给出李超代数恒等式新证。
AI 中文摘要
我们通过伴随顶点算子的乘积，在福克空间中引入了由划分参数化的三个向量族\(|\underline{\lambda}^{so}\rangle\)、\(|\underline{\lambda}^{sp}\rangle\)和\(|\underline{\lambda}^{o}\rangle\)。我们证明对偶真空向量的商空间由一类特殊划分索引的划分向量张成。这些划分索引向量还帮助我们以一种与特殊划分族相关的新方式推导B、C和D型的对偶Littlewood恒等式。作为应用，我们得到一种新的自由费米子构造，表明Rains和Betea引入的与对偶Littlewood恒等式相关的测度相对于某些显式相关核是行列式形式的。此外，我们通过在完全对偶福克空间中与单列划分（或广义划分\((0^m)\)）索引的元素计算内积，建立了一些在特定受限划分上求和的广义Littlewood恒等式。特别地，对于每个正整数\(n\)，我们得到了\((-n)\)-不对称划分的广义Littlewood恒等式。我们表明这些广义Littlewood恒等式包含几个著名的Littlewood型恒等式作为特殊情况。因此，我们也给出了李超代数广义Littlewood恒等式的新证明。
英文摘要
We introduce three families of vectors $|\underlineλ^{so}\rangle$, $|\underlineλ^{sp}\rangle$ and $|\underlineλ^{o}\rangle$ parametrized by partitions in the Fock space by using products of adjoint vertex operators. We show that the quotient space of the dual vacuum vector is spanned by the partition vectors indexed by a special family of partitions. The partition-indexed vectors also help us to derive the dual Littlewood identities of types B, C, and D in a new manner associated to the special family of partitions. As an application, we obtain a new free fermionic construction to show that the measures related to dual Littlewood identities introduced by Rains \cite[Section 7]{Rai2000} and Betea \cite[Section 3]{Be2020} are determinantal with repect to some explicit correlation kernels. Furthermore we establish a number of generalized Littlewood identities summed over certain restricted partitions by computing the inner products with elements indexed by one-column partitions {or generalized partitions $(0^m)$} in the complete dual Fock space. {In particular, for each positive integer $n$, we obtain generalized Littlewood identities for $(-n)$-asymmetric partitions. We show that these generalized Littlewood identities contain several well-known Littlewood-type identities as special cases. Consequently we also give a new proof of the generalized Littlewood identity \cite[(5.25)]{LSV2008} for Lie superalgebras. } %We also produce infinite generalized Littlewood identities by calculating the inner products between these elements and some elements indexed by one-column partitions in the complete dual Fock space.
Comments22pp
Journal refForum Math. Sigma 14 (2026), e106, 1-20
