Schur Q-functions Research Articles

KdV and mKdV Hierarchies and Schur Q-functions

KdV and mKdV Hierarchies and Schur Q-functions

Littlewood-Richardson rule for generalized Schur Q-functions

Littlewood-Richardson rule for generalized Schur Q-functions

Extending a word property for twisted Coxeter systems

We prove two extensions of Hansson and Hultman's word property for certain analogues of reduced words associated to twisted involutions in Coxeter groups. Our first extension concerns the superset of such words in which terms with a natural commutativity property may be optionally primed. Our other extension involves variants of these words in which a defining minimal length condition is relaxed. In type A the sets considered are closely related to generating functions for Schur Q-functions and K-theoretic Schur P-functions.

Quantum field presentation for generalized Hall–Littlewood functions

In this paper, we give the definition and quantum field presentation for a generalized Hall–Littlewood function using vertex operators, which is an extension of the Hall–Littlewood function. Furthermore, these generalized Hall–Littlewood functions can be deduced to universal characters at t = 0 and generalized Schur Q-functions at t = −1.

Multiparameter universal characters of B-type and integrable hierarchy

We introduce the multiparameter universal characters of B-type (B-type universal character), which contain special factorial Schur Q-functions, classical Schur Q-functions, and classical B-type universal characters. Then, we prove that multiparameter B-type universal characters are solutions of the universal character hierarchy of B-type.

Spin Hurwitz theory and Miwa transform for the Schur Q-functions

Schur functions are the common eigenfunctions of generalized cut-and-join operators which form a closed algebra. They can be expressed as differential operators in time-variables and also through the eigenvalues of auxiliary N×N matrices X, known as Miwa variables. Relevant for the cubic Kontsevich model and also for spin Hurwitz theory is an alternative set of Schur Q-functions. They appear in representation theory of the Sergeev group, which is a substitute of the symmetric group, related to the queer Lie superalgebras q(N). The corresponding spin Wˆ-operators were recently found in terms of time-derivatives, but a substitute of the Miwa parametrization remained unknown, which is an essential complication for the matrix model technique and further developments. We demonstrate that the Miwa representation, in this case, involves a fermionic matrix Ψ in addition to X, but its realization using supermatrices is not quite naive.

Intersection numbers on {overline{M}}_{g,n} and BKP hierarchy

In their recent inspiring paper, Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar description for the Brézin-Gross-Witten tau-function. Moreover, we identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy.

Bilinear expansion of Schur functions in Schur Q-functions: A fermionic approach

An identity is derived expressing Schur functions as sums over products of pairs of Schur Q Q -functions, generalizing previously known special cases. This is shown to follow from their representations as vacuum expectation values (VEV’s) of products of either charged or neutral fermionic creation and annihilation operators, Wick’s theorem and a factorization identity for VEV’s of products of two mutually anticommuting sets of neutral fermionic operators.

Symplectic Q-functions

Symplectic Q-functions are a symplectic analogue of Schur Q-functions and defined as the t=−1 specialization of Hall–Littlewood functions associated with the root system of type C. In this paper we prove that symplectic Q-functions share many of the properties of Schur Q-functions, such as a tableau description and a Pieri-type rule. And we present some positivity conjectures, including the positivity conjecture of structure constants for symplectic P-functions. We conclude by giving a tableau description of factorial symplectic Q-functions.

Bilinear expansions of lattices of KP τ -functions in BKP τ -functions: A fermionic approach

We derive a bilinear expansion expressing elements of a lattice of Kadomtsev-Petviashvili (KP) τ-functions, labeled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP τ-functions, labeled by strict partitions. This generalizes earlier results relating determinants and Pfaffians of minors of skew symmetric matrices, with applications to Schur functions and Schur Q-functions. It is deduced using the representations of KP and BKP τ-functions as vacuum expectation values (VEVs) of products of fermionic operators of charged and neutral type, respectively. The lattice is generated by the insertion of products of pairs of charged creation and annihilation operators. The result follows from expanding the product as a sum of monomials in the neutral fermionic generators and applying a factorization theorem for VEVs of products of operators in the mutually commuting subalgebras. Applications include the case of inhomogeneous polynomial τ-functions of KP and BKP type.

K-theory formulas for orthogonal and symplectic orbit closures

The complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits. We study two families of polynomials introduced by Wyser and Yong representing the K-theory classes of the closures of these orbits. Our polynomials are analogous to the Grothendieck polynomials representing K-classes of Schubert varieties, and we show that like Grothendieck polynomials, they are uniquely characterized among all polynomials representing the relevant classes by a certain stability property. We show that the same polynomials represent the equivariant K-classes of symmetric and skew-symmetric analogues of Knutson and Miller's matrix Schubert varieties. We derive explicit expressions for these polynomials in special cases, including a Pfaffian formula relying on a more general degeneracy locus formula of Anderson. Finally, we show that taking an appropriate limit of our representatives recovers the K-theoretic Schur Q-functions of Ikeda and Naruse.

An analogue of chromatic bases and p-positivity of skew Schur Q-functions

We investigate chromatic symmetric functions in the relation to the algebra $\Gamma$ of symmetric functions generated by Schur $Q$-functions. We construct natural bases of $\Gamma$ in terms of chromatic symmetric functions. We also consider the $p$-positivity of skew Schur $Q$-functions and find a class of $p$-positive ribbon Schur $Q$-functions, making a conjecture that they are \emph{all}. We include many concrete computational results that support our conjecture.

Pfaffian identities and Virasoro operators

A formula for Schur Q-functions is presented which describes the action of the Virasoro operators. For a strict partition $$\lambda =(\lambda _1,\lambda _2,\ldots ,\lambda _{2m})$$, we show that, for $$k\ge 1$$, $$L_{k}Q_{\lambda } = \sum ^{2m}_{i= 1}(\lambda _i-k)Q_{\lambda -2k\epsilon _i}$$, where $$L_k$$ is the Virasoro operator given as the quadratic form of free bosons. This main formula follows from the Plucker-like bilinear identity of Q-functions as Pfaffians: $$\sum ^{2m}_{i=2}(-1)^{i}\partial _1Q_{\lambda _1,\lambda _i}\partial _1Q_{\lambda _2, \ldots ,\widehat{\lambda _i},\ldots , \lambda _{2m}}=0$$, where $$\partial _1=\partial /\partial t_1$$. This bilinear identity must be explained in geometric words. We conjecture that the Hirota bilinear equations of the KdV hierarchy are derived from this bilinear identity.

Strongly Coupled B-Type Universal Characters and Hierarchies

We construct a solution expressed in terms of Schur Q-functions of a strongly coupled B-type Kadomtsev-Petviashvili hierarchy. As a generalization of these functions, we introduce universal characters satisfying the bilinear equations of a new infinite-dimensional integrable system called the strongly coupled B-type universal character hierarchy.

The center of the twisted Heisenberg category, factorial Schur Q-functions, and transition functions on the Schur graph

We establish an isomorphism between the center of the twisted Heisenberg category and the subalgebra \(\Gamma \) of the symmetric functions generated by odd power sums. We give a graphical description of the factorial Schur Q-functions and inhomogeneous power sums as closed diagrams in the twisted Heisenberg category and show that the bubble generators of the center correspond to two sets of generators of \(\Gamma \) which encode data related to up/down transition functions on the Schur graph. Finally, we describe an action of the trace of the twisted Heisenberg category, the W-algebra \(W^-\subset W_{1+\infty }\), on \(\Gamma \).

Pfaffian formulas and Schur Q-function identities

We establish Pfaffian analogues of the Cauchy–Binet formula and the Ishikawa–Wakayama minor-summation formula. Each of these Pfaffian analogues expresses a sum of products of subpfaffians of two skew-symmetric matrices in terms of a single Pfaffian. By using these Pfaffian formulas we give transparent proofs to several identities for Schur Q-functions.

Dual Multiparameter Schur Q-Functions

For the Schur Q-functions there is a Cauchy identity, which shows a duality between the Schur P- and Q-functions. We will be interested in the multiparameter Schur Q-functions, which were introduced by V. N. Ivanov, and we will give dual analogs of the multiparameter Schur Q(P)-functions, with a corresponding multiparameter Cauchy identity.

Toward a polynomial basis of the algebra of peak quasisymmetric functions

Hazewinkel proved the Ditters conjecture that the algebra of quasisymmetric functions over the integers is free commutative by constructing a nice polynomial basis. In this paper, we prove a structure theorem for the algebra of peak quasisymmetric functions (PQSym) over the integers. It provides a polynomial basis of PQSym over the rational field, different from Hsiao's basis, and implies the freeness of PQSym over its subring of symmetric functions spanned by Schur's Q-functions.

Representation theory of 0-Hecke–Clifford algebras

The representation theory of 0-Hecke–Clifford algebras as a degenerate case is not semisimple and also with rich combinatorial meaning. Bergeron, Hivert and Thibon have proved that the Grothendieck ring of the category of finitely generated supermodules of 0-Hecke–Clifford algebras is isomorphic to the algebra of peak quasisymmetric functions defined by Stembridge. In this paper we further study the category of finitely generated projective supermodules and clarify the correspondence between it and the peak algebra of symmetric groups. In particular, two kinds of restriction rules for induced projective supermodules are obtained. After that, we consider the corresponding Heisenberg double and its Fock representation to prove that the ring of peak quasisymmetric functions is free over its subring of symmetric functions spanned by Schur's Q-functions.

A lift of Schur's Q-functions to the peak algebra

We construct a lift of Schur's Q-functions to the peak algebra of the symmetric group, called the noncommutative Schur Q-functions, and extract from them a new natural basis with several nice properties such as the positive right-Pieri rule, combinatorial expansion, etc. Dually, we get a basis of the Stembridge algebra of peak functions refining Schur's P-functions in a simple way.