Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs

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 then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials

Lagrange inversion formula, Laguerre polynomials and the free unitary Brownian motion

This paper is devoted to the computations of some relevant quantities associated with the free unitary Brownian motion. Using the Lagrange inversion formula, we first derive an explicit expression for its alternating star cumulants of even lengths and relate them to those having odd lengths by means of a summation formula for the free cumulants with product as entries. Next, we use again this formula together with a generating series for Laguerre polynomials in order to compute the Taylor coefficients of the reciprocal of the $R$-transform of the free Jacobi process associated with a single projection of rank $1/2$ and those of the $S$-transform as well. This generating series lead also to the Taylor expansions of the Schur function of the spectral distribution of the free unitary Brownian motion and of its first iterate.Comment: last version: other typos are correcte