Related papers: Skew hook Schur functions and the cyclic sieving p…
We make a broad conjecture about the $k$-Schur positivity of Catalan functions, symmetric functions which generalize the (parabolic) Hall-Littlewood polynomials. We resolve the conjecture with positive combinatorial formulas in cases which…
Our contribution is two-folded. First, starting from the known fact that every real skew-Hamiltonian matrix has a real Hamiltonian square root, we give a complete characterization of the square roots of a real skew-Hamiltonian matrix W.…
Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and…
We resolve affirmatively some conjectures of Reiner, Stanton, and White \cite{ReinerComm} regarding enumeration of transportation matrices which are invariant under certain cyclic row and column rotations. Our results are phrased in terms…
Classical Schur P-functions are the particular case of Hall-Littlewood polynomials when the parameter is equal to -1. We introduce factorial (interpolation) analogues of Schur P-functions. A dimension of a skew shifted Young diagram is the…
The classical Littlewood-Richardson rule is a rule for computing coefficients in many areas, and comes in many guises. In this paper we prove two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions that are…
We give a Littlewood-Richardson type rule for expanding the product of a row-strict quasisymmetric Schur function and a symmetric Schur function in terms of row-strict quasisymmetric Schur functions. We then discuss a family of polynomials…
In this note, we provide a short proof of Theorem 3.3 in the paper titled \emph{Crystals, semistandard tableaux and cyclic sieving phenomenon}, by Y.-T.~Oh and E.~Park, which concerns a cyclic sieving phenomenon on semi-standard Young…
We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum…
The generic Hecke algebra for the hyperoctahedral group, i.e. the Weyl group of type B, contains the generic Hecke algebra for the symmetric group, i.e. the Weyl group of type A, as a subalgebra. Inducing the index representation of the…
We use Young's raising operators to give short and uniform proofs of several well known results about Schur polynomials and symmetric functions, starting from the Jacobi-Trudi identity.
We prove two conjectures of Shareshian and Wachs about Eulerian quasisymmetric functions and polynomials. The first states that the cycle type Eulerian quasisymmetric function $Q_{\lambda,j}$ is Schur-positive, and moreover that the…
Assuming that Siegel zeros exist, we prove a hybrid version of the Chowla and Hardy--Littlewood prime tuples conjectures. Thus, for an infinite sequence of natural numbers $x$, and any distinct integers $h_1,\dots,h_k,h'_1,\dots,h'_\ell$,…
The involution Stanley symmetric functions $\hat{F}_y$ are the stable limits of the analogues of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for…
The use of skew polynomial rings allows to endow linear codes with cyclic structures which are not cyclic in the classical (commutative) sense. Whenever these skew cyclic structures are carefully chosen, some control over the Hamming…
Skew stable Grothendieck polynomials are $K$-theoretic analogues of skew Schur polynomials. We give a free-fermionic presentation of skew stable Grothendieck polynomials and their dual symmetric functions. By using our presentation, we…
The number of standard Young tableaux of a skew shape $\lambda/\mu$ can be computed as a sum over excited diagrams inside $\lambda$. Excited diagrams are in bijection with certain lozenge tilings, with flagged semistandard tableaux and also…
We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t-analogs using vertex operators. We study subspaces forming a filtration for…
Orbit harmonics is a tool in combinatorial representation theory which promotes the (ungraded) action of a linear group $G$ on a finite set $X$ to a graded action of $G$ on a polynomial ring quotient by viewing $X$ as a $G$-stable point…
We obtain a new formula to relate the value of a Schur polynomial with variables $(x_1,\ldots,x_N)$ with values of Schur polynomials at $(1,\ldots,1)$. This allows to study the limit shape of perfect matchings on a square hexagon lattice…