3,522 research outputs found
The probability of positivity in symmetric and quasisymmetric functions
Given an element in a finite-dimensional real vector space, , that is a
nonnegative linear combination of basis vectors for some basis , we compute
the probability that it is furthermore a nonnegative linear combination of
basis vectors for a second basis, . We then apply this general result to
combinatorially compute the probability that a symmetric function is
Schur-positive (recovering the recent result of Bergeron--Patrias--Reiner),
-positive or -positive. Similarly we compute the probability that a
quasisymmetric function is quasisymmetric Schur-positive or
fundamental-positive. In every case we conclude that the probability tends to
zero as the degree of a function tends to infinity
Asymmetric function theory
The classical theory of symmetric functions has a central position in
algebraic combinatorics, bridging aspects of representation theory,
combinatorics, and enumerative geometry. More recently, this theory has been
fruitfully extended to the larger ring of quasisymmetric functions, with
corresponding applications. Here, we survey recent work extending this theory
further to general asymmetric polynomials.Comment: 36 pages, 8 figures, 1 table. Written for the proceedings of the
Schubert calculus conference in Guangzhou, Nov. 201
- …