The genomic Schur function is fundamental-positive

Abstract

In work with A. Yong, the author introduced genomic tableaux to prove the first positive combinatorial rule for the Littlewood-Richardson coefficients in torus-equivariant K-theory of Grassmannians. We then studied the genomic Schur function U λ , a generating function for such tableaux, showing that it is non-trivially a symmetric function, although generally not Schur-positive. Here we show that U λ is, however, positive in the basis of fundamental quasisymmetric functions. We give a positive combinatorial formula for this expansion in terms of gapless increasing tableaux; this is, moreover, the first finite expression for U λ . Combined with work of A. Garsia and J. Remmel, this yields a compact combinatorial (but necessarily non-positive) formula for the Schur expansion of U λ .

Asymmetric function theory

“…Alternatively, perhaps there is a general representation-theoretic construction that yields all of these various βdeformations. No such construction is currently known, but for some ideas along these lines see [Gal17,MPS18a,Pec18]. Table 1.…”