22,335 research outputs found
Differential posets have strict rank growth: a conjecture of Stanley
We establish strict growth for the rank function of an r-differential poset.
We do so by exploiting the representation theoretic techniques developed by
Reiner and the author for studying related Smith forms.Comment: 5 pages, 1 figur
Note on parity and the irreducible characters of the symmetric group
The object of this short note is to prove a theorem and present a conjecture
for the number of even entries in the character table of the symmetric group
The probability that a character value is zero for the symmetric group
We consider random character values X(g) of the symmetric group on n symbols,
where X is chosen at random from the set of irreducible characters and g is
chosen at random from the group, and we show that X(g)=0 with probability
tending to one as n tends to infinity.Comment: 3 page
Walls in Milnor fiber complexes
For a real reflection group the reflecting hyperplanes cut out on the unit
sphere a simplicial complex called the Coxeter complex. Abramenko showed that
each reflecting hyperplane meets the Coxeter complex in another Coxeter complex
if and only if the Coxeter diagram contains no subdiagram of type , ,
or . The present paper extends Abramenko's result to a wider class of
complex reflection groups. These groups have a Coxeter-like presentation and a
Coxeter-like complex called the Milnor fiber complex. Our first main theorem
classifies the groups whose reflecting hyperplanes meet the Milnor fiber
complex in another Milnor fiber complex. To understand better the walls that
fail to be Milnor fiber complexes we introduce Milnor walls. Our second main
theorem generalizes Abramenko's result in a second way. It says that each wall
of a Milnor fiber complex is a Milnor wall if and only if the diagram contains
no subdiagram of type , , or
Some stable homology calculations and Occam's razor for Hodge structures
Motivated by motivic zeta function calculations, Vakil and Wood in [VMW12]
made several conjectures regarding the topology of subspaces of symmetric
products. The purpose of this note is to prove two of these conjectures and
disprove a strengthening of one of them.Comment: 7 page
Homological stability for topological chiral homology of completions
By proving that several new complexes of embedded disks are highly connected,
we obtain several new homological stability results. Our main result is
homological stability for topological chiral homology on an open manifold with
coefficients in certain partial framed -algebras. Using this, we prove a
special case of a conjecture of Vakil and Wood on homological stability for
complements of closures of particular strata in the symmetric powers of an open
manifold and we prove that the bounded symmetric powers of closed manifolds
satisfy homological stability rationally.Comment: 51 pages, 7 figures, major revision. To appear in Advances in
Mathematic
Differential posets and Smith normal forms
We conjecture a strong property for the up and down maps U and D in an
r-differential poset: DU+tI and UD+tI have Smith normal forms over Z[t]. In
particular, this would determine the integral structure of the maps U, D, UD,
DU, including their ranks in any characteristic.
As evidence, we prove the conjecture for the Young-Fibonacci lattice YF
studied by Okada and its r-differential generalizations Z(r), as well as
verifying many of its consequences for Young's lattice Y and the r-differential
Cartesian products Y^r.Comment: 29 pages, 9 figure
Homological stability for complements of closures
We prove Conjecture F from [VW12] which states that the complements of
closures of certain strata of the symmetric power of a smooth irreducible
complex variety exhibit rational homological stability. Moreover, we generalize
this conjecture to the case of connected manifolds of dimension at least 2 and
give an explicit homological stability range.Comment: 15 page
Sharper periodicity and stabilization maps for configuration spaces of closed manifolds
In this note we study the homology of configuration spaces of closed
manifolds. We sharpen the eventual periodicity results of Nagpal and
Cantero-Palmer, prove integral homological stability for configuration spaces
of odd-dimensional manifolds and introduce a stabilization map on the homology
with -coefficients of configuration spaces of odd-dimensional
manifolds.Comment: 9 pages, 1 figure. Improved the result for odd-dimensional manifolds.
To appear in Proceedings of the AM
Improved homological stability for configuration spaces after inverting 2
In Appendix A of his article on rational functions, Segal proved homological
stability for configuration spaces with a stability slope of 1/2. This was
later improved to a slope of 1 by Church and Randal-Williams if one works with
rational coefficients and manifolds of dimension at least . In this note we
prove that the stability slope of 1 holds even with Z[1/2] coefficients, and
clarify some aspects of Segal's proof for topological manifolds.Comment: 11 pages, 1 figure. Minor mistakes fixe
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