Related papers: Open Questions for operators related to Rectangula…
We consider families of quasisymmetric functions with the property that if a symmetric function $f$ is a positive sum of functions in one of these families, then f is necessarily a positive sum of Schur functions. Furthermore, in each of…
The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a ``raising operator" and Fock space construction. A simple graphical proof of this relation is proposed. The new operator…
We consider a filtration of the symmetric function space given by $\Lambda^{(k)}_t$, the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than $k$. We introduce symmetric functions called the…
We construct a new class of operators that act on symmetric functions with two deformation parameters $q$ and $t$. Our combinatorial construction associates each operator with a specific lattice path, whose steps alternate between moving up…
Our main result here is that the specialization at $t=1/q$ of the $Q_{km,kn}$ operators studied in [4] may be given a very simple plethystic form. This discovery yields elementary and direct derivations of several identities relating these…
The action of the Bernstein operators on Schur functions was given in terms of codes in [CG] and extended to the analog in Schur Q-functions in [HJS]. We define a new combinatorial model of extended codes and show that both of these results…
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 derive an explicit simple formula for expectations of all Schur functions in the real Ginibre ensemble. It is a positive integer for all entries of the partition even and zero otherwise. The result can be used to determine the average of…
FPSAC 2013 Extended Abstract. We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand…
A new construction for the form sum of positive, selfadjoint operators is given in this paper. The situation is a bit more general, because our aim is to add positive, symmetric operators. With the help of the used method, some commutation…
Recent advances in the theory of complex symmetric operators are presented and related to current studies in non-hermitian quantum mechanics. The main themes of the survey are: the structure of complex symmetric operators, $C$-selfadjoint…
We generalize several classical results about Schur functions to the family of cylindric Schur functions. First, we give a combinatorial proof of a Murnaghan--Nakayama formula for expanding cylindric Schur functions in the power-sum basis.…
Given an element in a finite-dimensional real vector space, $V$, that is a nonnegative linear combination of basis vectors for some basis $B$, we compute the probability that it is furthermore a nonnegative linear combination of basis…
We show that some matrices are Schur multipliers and this is applied to obtain classes of operator-valued Foguel-Hankel operators similar to contractions. This provides partial answers to a problem of K. Davidson and the second author…
For operators on Hilbert spaces of any dimension, we show that equivalence after extension coincides with equivalence after one-sided extension, thus obtaining a proof of their coincidence with Schur coupling. We also provide a concrete…
This is a short note about Schur positivity. We introduce Schur polynomials and explain how they appear in the representation theory of the general linear group. We end with a new result of the author with F. Bergeron and V. Reiner that…
In this paper we give the answers to two open questions on complex symmetric composition operators. By doing this, we give a complete description of complex symmetric composition operators whose symbols are linear fractional.
We prove Okounkov's conjecture, a conjecture of Fomin-Fulton-Li-Poon, and a special case of Lascoux-Leclerc-Thibon's conjecture on Schur positivity and give several more general statements using a recent result of Rhoades and Skandera. An…
We determine the precise conditions under which any skew Schur function is equal to a Schur function over both infinitely and finitely many variables.
Let $\Lambda$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_\lambda[X/(1-t)]\}_{\lambda_1\leq k}$. We introduce a new family of symmetric polynomials,…