Standard Parabolic Subgroup Research Articles

Bar operators for quasiparabolic conjugacy classes in a Coxeter group

The action of a Coxeter group W on the set of left cosets of a standard parabolic subgroup deforms to define a module MJ of the group's Iwahori–Hecke algebra H with a particularly simple form. Rains and Vazirani have introduced the notion of a quasiparabolic set to characterize W-sets for which analogous deformations exist; a motivating example is the conjugacy class of fixed-point-free involutions in the symmetric group. Deodhar has shown that the module MJ possesses a certain antilinear involution, called the bar operator, and a certain basis invariant under this involution, which generalizes the Kazhdan–Lusztig basis of H. The well-known significance of this basis in representation theory makes it natural to seek to extend Deodhar's results to the quasiparabolic setting. In general, the obstruction to finding such an extension is the existence of an appropriate quasiparabolic analogue of the “bar operator.” In this paper, we consider the most natural definition of a quasiparabolic bar operator, and develop a theory of “quasiparabolic Kazhdan–Lusztig bases” under the hypothesis that such a bar operator exists. Giving content to this theory, we prove that a bar operator in the desired sense does exist for quasiparabolic W-sets given by twisted conjugacy classes of twisted involutions. Finally, we prove several results classifying the quasiparabolic conjugacy classes in a Coxeter group.

On double cosets with the trivial intersection property and Kazhdan-Lusztig cells in $S_n$

‎For a composition $lambda$ of $n$ our aim is to obtain reduced forms‎ ‎for all the elements in the ‎$w_{J(lambda)}$‎, ‎the longest element of the standard parabolic‎ ‎subgroup of $S_n$ corresponding to $lambda$‎. ‎We investigate how far this is possible to achieve by looking at‎ ‎elements of the form $w_{J(lambda)}d$‎, ‎where $d$ is a prefix of‎ ‎an element of minimum length in a $(W_{J(lambda)},B)$ double coset‎ ‎with the trivial intersection property‎, ‎$B$ being a parabolic subgroup‎ ‎of $S_n$ whose type is `dual' to that of $W_{J(lambda)}$‎.

Convexity of parabolic subgroups in Artin groups

We prove that any standard parabolic subgroup of any Artin group is convex with respect to the standard generating set.

On excess in finite Coxeter groups

For a finite Coxeter group W and w an element of W the excess of w is defined to be e(w)=min⁡{ℓ(x)+ℓ(y)−ℓ(w)|w=xy,x2=y2=1} where ℓ is the length function on W. Here we investigate the behaviour of e(w), and a related concept reflection excess, when restricted to standard parabolic subgroups of W. Also the set of involutions inverting w is studied.

Parabolic Subgroups of SO2l Over a Dedekind Ring of Arithmetic Type

Let R be a commutative ring all of whose proper factor rings are finite and such that there exists a unit of infinite order. We show that for a subgroup P in G = SO(2l, R), l ≥ 3, containing the Borel subgroup B, the following alternative holds: either P contains a relative elementary subgroup E I for some ideal I = 0 or H is contained in a proper standard parabolic subgroup. For Dedekind rings of arithmetic type this allows us, under some mild additional assumptions on units, to describe completely the overgroups of B in G. Earlier, similar results for the special linear and symplectiv groups were obtained by A. V. Alexandrov and the second author. The proofs in the present paper follow the same general strategy, but are noticeably harder, from a technical viewpoint. Bibliography: 34 titles.

Deformations of permutation representations of Coxeter groups

The permutation representation afforded by a Coxeter group W acting on the cosets of a standard parabolic subgroup inherits many nice properties from W such as a shellable Bruhat order and a flat deformation over ?[q] to a representation of the corresponding Hecke algebra. In this paper we define a larger class of subgroups (more generally, quasiparabolic W-sets), and show that they also inherit these properties. Our motivating example is the action of the symmetric group on fixed-point-free involutions by conjugation.

A note on element centralizers in finite Coxeter groups

Abstract The normalizer NW (WJ ) of a standard parabolic subgroup WJ of a finite Coxeter group W splits over the parabolic subgroup with complement NJ consisting of certain minimal length coset representatives of WJ in W. In this note we show that (with the exception of a small number of cases arising from a situation in Coxeter groups of type Dn ) the centralizer CW (w) of an element w ∈ W is in a similar way a semidirect product of the centralizer of w in a suitable small parabolic subgroup WJ with complement isomorphic to the normalizer complement NJ . Then we use this result to give a new short proof of Solomon's Character Formula and discuss its connection to MacMahon's master theorem.

Hom-configurations and noncrossing partitions

We study maximal Hom-free sets in the ?[2]-orbit category C(Q) of the bounded derived category for the path algebra associated to a Dynkin quiver Q, where ? denotes the Auslander---Reiten translation and [2] denotes the square of the shift functor. We prove that these sets are in bijection with periodic combinatorial configurations, as introduced by Riedtmann, certain Hom??0-configurations, studied by Buan, Reiten and Thomas, and noncrossing partitions of the Coxeter group associated to Q which are not contained in any proper standard parabolic subgroup. Note that Reading has proved that these noncrossing partitions are in bijection with positive clusters in the associated cluster algebra. Finally, we give a definition of mutation of maximal Hom-free sets in $\mathcal {C}(Q)$ and prove that the graph of these mutations is connected.

Parabolic subgroups of SL n and Sp2l over a Dedekind ring of arithmetic type

Let R be a commutative ring all of whose proper factor rings are finite and in which a unit of infinite order exists. It is shown that for a subgroup P in G =SL(n, R), n ≥3,or in G =Sp(2l, R), l ≥2, containing the standard Borel subgroup B, the following alternative holds: either P contains a relative elementary subgroup EI for some ideal I≠0 or H is contained in a proper standard parabolic subgroup. For Dedekind rings of arithmetic type, this allows one, under some mild additional assumptions on units, to completely describe the overgroups of B in G. Bibliography: 30 titles.

On parabolic closures in Coxeter groups

For a finitely generated subgroup W′ of a Coxeter system (W, S), there are finitely generated reflection subgroups W1, . . . , Wn of W, each containing W′, such that any reflection subgroup of W containing W′ contains one of the Wi as a standard parabolic subgroup. The canonical Coxeter generators of the Wi , and an expression for the parabolic closure of W′ as a W-conjugate of a standard parabolic subgroup of W, may be effectively determined.

Subgroup conjugacy problem for Garside subgroups of Garside groups

We solve the subgroup conjugacy problem for parabolic subgroups and Garside subgroups of a Garside group, and we present deterministic algorithms. This solution may be improved by using minimal simple elements. For standard parabolic subgroups of Garside groups we provide effective algorithms for computing minimal simple elements.

Volume of truncated fundamental domains

For T > 0, by applying Arthur's analytic truncation AT to the constant function 1 on G(F)\G(A)1, we obtain a characteristic function of a compact subset i(T) of G(F)\G(A)1. The aim of this paper is to give a general formula for the volume of a(T), which is an exponential polynomial function of T. Consequently, by letting T -* oo, we obtain a new way to evaluate the volume of fundamental domain for G(F)\G(A)1. While a(T) seems to be quite arbitrary, its volume admits beautiful structures: (i) Surprisingly enough, the coefficients of T are special values of zeta functions. (ii) The volume is an alternating sum associated to standard parabolic subgroups. ((ii) is suggested by geometry after all, a(T) is obtained from G(F)\G(A)1 by truncating off cuspidal regions which are known to be parametrized by standard parabolic subgroups.) Besides all basic properties of Arthur's analytic truncation, our method is based on a fundamental formula of Jacquet-Lapid-Rogawski on an integration of truncated Eisenstein series associated to cusp forms. Their formula, an advanced version of the Rankin-Selberg method, is obtained using the so-called Bernstein principle via regularized integration over cones.

On parabolic submonoids of a class of singular Artin monoids

AbstractThis paper concerns parabolic submonoids of a class of monoids known as singular Artin monoids. The latter class includes the singular braid monoid— a geometric extension of the braid group, which was created for the sole purpose of studying Vassiliev invariants in knot theory. However, those monoids may also be construed (and indeed, are defined) as a formal extension of Artin groups which, in turn, naturally generalise braid groups. It is the case, by van der Lek and Paris, that standard parabolic subgroups of Artin groups are canonically isomorphic to Artin groups. This naturally invites us to consider whether the same holds for parabolic submonoids of singular Artin monoids. We show that it is in fact true when the corresponding Coxeter matrix is of ‘type FC’ hence generalising Corran's result in the ‘finite type’ case.

Equivalence and congruence of matrices under the action of standard parabolic subgroups

Necessary and sufficient conditions for the equivalence and congruence of matrices under the action of standard parabolic subgroups are discussed.

Quantization of branching coefficients for classical Lie groups

We study natural quantizations K of branching coefficients corresponding to the restrictions of the classical Lie groups to the Levi subgroups of their standard parabolic subgroups. The polynomials obtained can be regarded as generalizations of Lusztig q-analogues of weight multiplicities. For GL n they coincide with Poincaré polynomials previously studied by Shimozono and Weyman. They also appear in the Hilbert series of the Euler characteristic of certain graded virtual G-modules and, by a result of Broer, admit nonnegative coefficients providing that restrictive conditions are verified. When the Levi subgroup considered is isomorphic to a direct product of linear groups, we prove that these polynomials admit a stable limit K ˜ which decomposes as nonnegative integer combination of Poincaré polynomials. For a general Levi subgroup, it is conjectured that the polynomials K have nonnegative coefficients when they are parametrized by two partitions. When G = GL n , the polynomials K can be interpreted as quantizations of the Littlewood–Richardson coefficients. We show that there also exists a duality between tensor product coefficients for types B, C, D (defined as the analogues of the Littlewood–Richardson coefficients) and branching coefficients corresponding to the restriction of SO 2 n to subgroups defined from orthogonal decompositions of the root system D n (which are not Levi subgroups). These coefficients can also be quantified but the q-analogues obtained may have negative coefficients. Given a tensor product Π of irreducible GL N -modules, we then introduce for each classical group G = SO N or Sp N some q-analogues D of the multiplicities obtained by decomposing Π into its G-irreducible components. We establish a duality between the polynomials D and K ˜ . According to a conjecture by Shimozono, the stable one-dimensional sums for nonexceptional affine crystals are expected to coincide with the polynomials D associated to a sequence of rectangular partitions of decreasing heights.

Artin–Tits groups with [formula omitted] Deligne complex

Let ( A , S ) be an Artin–Tits system, and A X be the standard parabolic subgroup of A generated by a subset X of S . Under the hypothesis that the Deligne complex has a CAT ( 0 ) geometric realization, we prove that the normalizer and the commensurator of A X in A are equal. Furthermore, if A X is of spherical type, these subgroups are the product of A X with the quasi-centralizer of A X in A . For two-dimensional Artin–Tits groups, the result still holds without any sphericality hypothesis on X . We explicitly describe the elements of this quasi-centralizer.

On Normal Subgroups of Coxeter Groups Generated by Standard Parabolic Subgroups

We discuss one construction of nonstandard subgroups in the category of Coxeter groups. Two formulae for the growth series of such a subgroups are given. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots.

A Combinatorial Construction of the Weak Order of a Coxeter Group

ABSTRACT Let W be a (finite or infinite) Coxeter group and W X be a proper standard parabolic subgroup of W. We show that the semilattice made up by W equipped with the weak order is a semidirect product of two smaller semilattices associated with W X . #Communicated by J. Aleu.

MATRIX COEFFICIENTS OF REPRESENTATIONS OF SU(2,2): THE CASE OF PJ-PRINCIPAL SERIES

The real Lie group G=SU(2,2) has a standard maximal parabolic subgroup PJ corresponding to the long simple roots in its restricted root system. We have an explicit formula of the holonomic system of the radial part of the matrix coefficients with corner K-type of the generalized principal series representations of G obtained by parabolic induction with respect to PJ from the discrete series representations of MJ. It is equivalent to the modified F2 system of Appell.

Parabolic subgroups of Artin groups of type FC

Let (A, S) be an Artin group of type FC and AT a standard parabolic subgroup of A. We use combinatorial tools to show that the normalizer of AT, the commensurator of AT, and the product of the quasi-centralizer of AT by AT are equal. Furthermore, we show that the centralizer and the quasi-centralizer of AT in A are generated by their intersections with the monoid A + .