196 research outputs found
Crystal energy functions via the charge in types A and C
The Ram-Yip formula for Macdonald polynomials (at t=0) provides a statistic
which we call charge. In types A and C it can be defined on tensor products of
Kashiwara-Nakashima single column crystals. In this paper we prove that the
charge is equal to the (negative of the) energy function on affine crystals.
The algorithm for computing charge is much simpler and can be more efficiently
computed than the recursive definition of energy in terms of the combinatorial
R-matrix.Comment: 25 pages; 1 figur
Signed Mahonians
A classical result of MacMahon gives a simple product formula for the
generating function of major index over the symmetric group. A similar
factorial-type product formula for the generating function of major index
together with sign was given by Gessel and Simion. Several extensions are given
in this paper, including a recurrence formula, a specialization at roots of
unity and type analogues.Comment: 23 page
Stable multivariate -Eulerian polynomials
We prove a multivariate strengthening of Brenti's result that every root of
the Eulerian polynomial of type is real. Our proof combines a refinement of
the descent statistic for signed permutations with the notion of real
stability-a generalization of real-rootedness to polynomials in multiple
variables. The key is that our refined multivariate Eulerian polynomials
satisfy a recurrence given by a stability-preserving linear operator. Our
results extend naturally to colored permutations, and we also give stable
generalizations of recent real-rootedness results due to Dilks, Petersen, and
Stembridge on affine Eulerian polynomials of types and . Finally,
although we are not able to settle Brenti's real-rootedness conjecture for
Eulerian polynomials of type , nor prove a companion conjecture of Dilks,
Petersen, and Stembridge for affine Eulerian polynomials of types and ,
we indicate some methods of attack and pose some related open problems.Comment: 17 pages. To appear in J. Combin. Theory Ser.
ENUMERATING PROJECTIVE REFLECTION GROUPS
Projective re ection groups have been recently dened by the second author. They include a special class of groups denoted G(r; p; s; n) which contains all classical Weyl groups and more generally all the complex re ection groups of type G(r; p; n). In this paper we dene some statistics analogous to descent number and major index over the projective re ection groups G(r; p; s; n), and we compute several generating functions concerning these parameters. Some aspects of the representation theory of G(r; p; s; n), as distribution of one-dimensional characters and computation of Hilbert series of invariant algebras, are also treated
Poset pinball, GKM-compatible subspaces, and Hessenberg varieties
This paper has three main goals. First, we set up a general framework to
address the problem of constructing module bases for the equivariant cohomology
of certain subspaces of GKM spaces. To this end we introduce the notion of a
GKM-compatible subspace of an ambient GKM space. We also discuss
poset-upper-triangularity, a key combinatorial notion in both GKM theory and
more generally in localization theory in equivariant cohomology. With a view
toward other applications, we present parts of our setup in a general algebraic
and combinatorial framework. Second, motivated by our central problem of
building module bases, we introduce a combinatorial game which we dub poset
pinball and illustrate with several examples. Finally, as first applications,
we apply the perspective of GKM-compatible subspaces and poset pinball to
construct explicit and computationally convenient module bases for the
-equivariant cohomology of all Peterson varieties of classical Lie type,
and subregular Springer varieties of Lie type . In addition, in the Springer
case we use our module basis to lift the classical Springer representation on
the ordinary cohomology of subregular Springer varieties to -equivariant
cohomology in Lie type .Comment: 32 pages, 4 figure
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