695 research outputs found
Some applications of Rees products of posets to equivariant gamma-positivity
The Rees product of partially ordered sets was introduced by Bj\"orner and
Welker. Using the theory of lexicographic shellability, Linusson, Shareshian
and Wachs proved formulas, of significance in the theory of gamma-positivity,
for the dimension of the homology of the Rees product of a graded poset
with a certain -analogue of the chain of the same length as . Equivariant
generalizations of these formulas are proven in this paper, when a group of
automorphisms acts on , and are applied to establish the Schur
gamma-positivity of certain symmetric functions arising in algebraic and
geometric combinatorics.Comment: Final version, with a section on type B Coxeter complexes added; to
appear in Algebraic Combinatoric
A q-analogue and a symmetric function analogue of a result by Carlitz, Scoville and Vaughan
We derive an equation that is analogous to a well-known symmetric function
identity: . Here the elementary symmetric
function is the Frobenius characteristic of the representation of
on the top homology of the subset lattice , whereas our
identity involves the representation of on
the Segre product of with itself. We then obtain a q-analogue of a
polynomial identity given by Carlitz, Scoville and Vaughan through examining
the Segre product of the subspace lattice with itself. We recognize
the connection between the Euler characteristic of the Segre product of
with itself and the representation on the Segre product of with
itself by recovering our polynomial identity from specializing the identity on
the representation of
On modular homology in projective space
AbstractFor a vector space V over GF(q) let Lk be the collection of subspaces of dimension k. When R is a field let Mk be the vector space over it with basis Lk. The inclusion map ∂:Mk→Mk−1 then is the linear map defined on this basis via ∂(X)≔∑Y where the sum runs over all subspaces of co-dimension 1 in X. This gives rise to a sequenceM:0←M0←M1←⋯←Mk−1←Mk←⋯which has interesting homological properties if R has characteristic p>0 not dividing q. Following on from earlier papers we introduce the notion of π-homological, π-exact and almost π-exact sequences where π=π(p,q) is some elementary function of the two characteristics. We show that M and many other sequences derived from it are almost π-exact. From this one also obtains an explicit formula for the Brauer character on the homology modules derived from M. For infinite-dimensional spaces we give a general construction which yields π-exact sequences for finitary ideals in the group ring RPΓL(V)
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