Measures and generalizations of dual Littlewood identities

Abstract

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.