18 research outputs found
Artin group injection in the Hecke algebra for right-angled groups
For any Coxeter system we consider the algebra generated by the projections
over the parabolic quotients. In the finite case it turn out that this algebra
is isomorphic to the monoid algebra of the Coxeter monoid (0-Hecke algebra). In
the infinite case it contains the Coxeter monoid algebra as a proper
subalgebra. This construction provides a faithful integral representation of
the Coxeter monoid algebra of any Coxeter system. As an application we will
prove that a right-angled Artin group injects in Hecke algebra of the
corresponding right-angled Coxeter group
The generalized lifting property of Bruhat intervlas
In [E. Tsukerman and L. Williams, {\em Bruhat Interval Polytopes}, Advances
in Mathematics, 285 (2015), 766-810] it is shown that every Bruhat interval of
the symmetric group satisfies the so-called generalized lifting property. In
this paper we show that a Coxeter group satisfies this property if and only if
it is finite and simply-laced.Comment: 18 page
Parabolically induced functions and equidistributed pairs
Given a function defined over a parabolic subgroup of a Coxeter group,
equidistributed with the length, we give a procedure to construct a function
over the entire group, equidistributed with the length. Such a procedure
permits to define functions equidistributed with the length in all the finite
Coxeter groups. We can establish our results in the general setting of graded
posets which satisfy some properties. These results apply to some known
functions arising in Coxeter groups as the major index, the negative major
index and the D-negative major index defined in type , and
respectively
Immanant varieties
We introduce immanant varieties, associated to simple characters of a finite
group. They include well studied classes of varieties, as Segre embeddings,
Grassmannians and some other Chow varieties. For a one-dimensional character
, we define -matroids by a maximality property. For trivial
characters, by exploring the combinatorics of incidence stratifications, we
provide a basis for the Chow vector spaces of the corresponding immanant
varieties
Wachs permutations, Bruhat order and weak order
We study the partial orders induced on Wachs and signed Wachs permutations by
the Bruhat and weak orders of the symmetric and hyperoctahedral groups. We show
that these orders are graded, determine their rank function, characterize their
ordering and covering relations, and compute their characteristic polynomials,
when partially ordered by Bruhat order, and determine their structure
explicitly when partially ordered by right weak order
Linear extensions and shelling orders
We prove that linear extensions of the Bruhat order of a matroid are shelling
orders and that the barycentric subdivision of a matroid is a Coxeter matroid,
viewing barycentric subdivisions as subsets of a parabolic quotient of a
symmetric group. A similar result holds for order ideals in minuscule quotients
of symmetric groups and in their barycentric subdivisions. Moreover, we apply
promotion and evacuation for labeled graphs of Malvenuto and Reutenauer to dual
graphs of simplicial complexes, providing promotion and evacuation of shelling
orders
-Jones-Wenzl idempotents
For a prime number and any natural number we introduce, by giving an
explicit recursive formula, the -Jones-Wenzl projector
, an element of the Temperley-Lieb algebra
with coefficients in . We prove that these projectors give the
indecomposable objects in the -Hecke category over , or equivalently, they give the projector in
to the top tilting module. The way in which we find these
projectors is by categorifying the fractal appearing in the expression of the
-canonical basis in terms of the Kazhdan-Lusztig basis for .Comment: 15 pages, 21 figures. Many minor changes. Major change of notation.
Final versio
The generalized lifting property of Bruhat intervals
In Tsukerman and Williams (Adv Math 285: 766\u2013810, 2015), it is shown that every Bruhat interval of the symmetric group satisfies the so-called generalized lifting property. In this paper, we show that a Coxeter group satisfies this property if and only if it is finite and simply-laced