Symmetric and quasi-symmetric functions associated to polymatroids

Abstract

To every subspace arrangement X we will associate symmetric functions P [X] and H [X]. These symmetric functions encode the Hilbert series and the minimal projective resolution of the product ideal associated to the subspace arrangement. They can be defined for discrete polymatroids as well. The invariant H[X] specializes to the Tutte polynomial T [X]. Billera, Jia and Reiner recently introduced a quasisymmetric function F [X] (for matroids) which behaves valuatively with respect to matroid base polytope decompositions. We will define a quasi-symmetric function G[X] for polymatroids which has this property as well. Moreover, G[X] specializes to P[X], H[X], T [X] and F [X].

Valuative invariants for polymatroids

“…The following theorem proves a conjecture of the first author in [7]: From G one can also construct a universal invariant for the covaluative property which specializes to Speyer's invariant.…”

“…However, it does not specialize to the Tutte polynomial. The first author introduced in [7] another quasi-symmetric function G. For some choice of basis {U α } of the ring of quasi-symmetric functions, G is defined by…”

Algebraic and Geometric Methods in Enumerative Combinatorics

“…Many other natural matroid functions were later discovered to be valuative. [AFR10,BJR09,Der09]. An example of a very general valuation on matroid polytopes from [AFR10] is the formal sum R(M ) = A⊆E R A,r(A) of symbols of the form R S,k where S is a subset of E and k is an integer.…”

“…In fact, the matroid M can clearly be recovered from R(M ). 18 Eventually, Derksen [Der09] constructed a valuative matroid invariant, which he and Fink proved to be universal. [DF10] We call it the Derksen-Fink invariant.…”

Recent Trends in Quasisymmetric Functions

“…Malvenuto and Reutenauer [90], through the Hopf algebraic dual, NSym, of QSym, introduce a quasisymmetric analogue of the power sum symmetric functions, also obtained independently by Derksen [29] using a similar process but with a computational error which leads to a different formula. To understand their construction, we recall several facts about generating functions for symmetric and noncommutative symmetric functions.…”