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Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras
... is known to be isomorphic to [ ], the group algebra of over ; see [39,Theorem 7.4.6]. In particular, they have the same representation theory: Every -module of finite -dimension is a direct sum of simple -modules, and the simple -modules are in bijection with the simple [ ]-modules. ...
... By (5.3), the bases ( −1 ) ∈ and ( ( ) −1 −1 ) ∈ are dual to each other with respect to − . Applying the second displayed equation on page 226 of [39], we see that the above sum simplifies to ...
... (In the notation of [39], we are taking to be the operator by which acts on the -module of character .) By (6.7), (6.4), and (6.15), ...
We solve two open problems in Coxeter–Catalan combinatorics. First, we introduce a family of rational noncrossing objects for any finite Coxeter group, using the combinatorics of distinguished subwords. Second, we give a type‐uniform proof that these noncrossing Catalan objects are counted by the rational Coxeter–Catalan number, using the character theory of the associated Hecke algebra and the properties of Lusztig's exotic Fourier transform. We solve the same problems for rational noncrossing parking objects.
... Proof. We argue by induction on n, starting with n = 4. Employing Algorithm B described in [3, §2.1] gives [4], [4,2], [4,2,1], [4,2,3], [4,2,1,3], [4,2,1,3,2], [4, 2, 1, 3, 2, 4]}. ...
... Proof. We argue by induction on n, starting with n = 4. Employing Algorithm B described in [3, §2.1] gives [4], [4,2], [4,2,1], [4,2,3], [4,2,1,3], [4,2,1,3,2], [4, 2, 1, 3, 2, 4]}. ...
... Proof. We argue by induction on n, starting with n = 4. Employing Algorithm B described in [3, §2.1] gives [4], [4,2], [4,2,1], [4,2,3], [4,2,1,3], [4,2,1,3,2], [4, 2, 1, 3, 2, 4]}. ...
For a Coxeter group W with length function , the excess zero graph has vertex set the non-identity involutions of W, with two involutions x and y adjacent whenever . Properties of this graph such as connectivity, diameter and valencies of certain vertices of are explored.
... The top row consists in applying Ψ CF , which by Proposition 2.11 is (F )•Ψ C •(F −1 ) to the first two legs, followed by the action BF = B •(F −1 ) on V . By (8), this gives the action of B F on Ind CF BF (V F ). The bottom row is the F -twisted action of B on Ind C B (V ). ...
... Proof. By [8,Section 5.5.4], the irreducible characters of B n are given as ...
... whereχ µ ∈ Irr(B a ) is the pullback of the irreducible character χ λ ∈ S a (see [8,Definition 5.4.4]) along the projection map B a → S a , ⊠ denotes outer tensor product, and ǫ ′ b is the restriction to B b of the linear character ǫ ′ on B n which sends t i → −1 and s i → +1. ...
Drinfeld twists, and the twists of Giaquinto and Zhang, allow for algebras and their modules to be deformed by a cocycle. We prove general results about cocycle twists of algebra factorisations and induced representations and apply them to reflection groups and rational Cherednik algebras. In particular, we describe how a twist acts on characters of Coxeter groups of type and and relate them to characters of mystic reflection groups. This is used to characterise twists of standard modules of rational Cherednik algebras as standard modules for certain braided Cherednik algebras. We introduce the coinvariant algebra of a mystic reflection group and use a twist to show that an analogue of Chevalley's theorem holds for these noncommutative algebras. We also discuss several cases where the negative braided Cherednik algebras are, and are not, isomorphic to rational Cherednik algebras.
... As in [2], we write SL n (q), SU n (q 1 2 ), SP n (q) and Ω ± n (q) for the finite classical groups acting naturally on the vector space F n q . As in [4] and [3] for type A, the irreducible representations of the Iwahori-Hecke algebras in types B and D are explicitly described by the Hoefsmit model (see [8] or [12]). The irreducible representations of the Iwahori-Hecke algebras are indexed by double-partitions λ of n and have a basis formed by the standard double-tableaux T associated with those double-partitions. ...
... In Section 2.1 we give the irreducible representations described by the Hoefsmit model in [8] and [12] (Theorem 2.1) and define a weight on standard tableaux and double-partitions of n which allows us to define a bilinear form verifying nice properties (Proposition 2.1). ...
... We set in this section F q = F p (α). As in [8], we take for the Iwahori-Hecke algebra of type D H Dn,α the sub-algebra of H Bn,α,1 generated by U = T S 1 T, S 1 , . . . S n−1 . ...
We determine the image of the Artin groups of types B and D inside the Iwahori-Hecke algebras, when defined over finite fields, in the semisimple case. This generalizes earlier work on type A by Brunat, Magaard and Marin. In this multi-parameter case, this image depends heavily on the parameters.
... The algebra H W2 is an Iwahori-Hecke algebra and its modular representation theory has been studied in [10]. Hence, from now on only the cases k = 3, 4, 5 will be considered. ...
... , which is split semisimple (see [11], theorem 5.2). Hence, by Tits' deformation theorem (see Theorem 7.4.6 in [10]) the specialization v i → 1 induces a bijection Irr(C(v)H W k ) → Irr(W k ), χ k → χ. By Theorem 7.2.6 in [10] we have: ...
... Hence, by Tits' deformation theorem (see Theorem 7.4.6 in [10]) the specialization v i → 1 induces a bijection Irr(C(v)H W k ) → Irr(W k ), χ k → χ. By Theorem 7.2.6 in [10] we have: ...
We classify all the decomposition matrices of the generic Hecke algebras on 3 strands in characteristic 0. These are the generic Hecke algebras associated to the exceptional complex reflection groups , and . We prove that for every choice of the parameters that define these algebras, all ordinary representations are obtained as modular reductions of irreducible representations.
... We write λ n to say that λ is a bipartition of n, and the set of bipartitions of n is denoted by Bip(n). It is well-known that the conjugacy classes of W n are in bijection with Bip(n) (see [9,14]). We defineλ as the signed composition of n obtained by concatenation of λ + and −λ − . ...
... In particular, w n w C is the longest element of X C (see [9]); ...
... This follows immediately from Proposition 4.23 and from the commutativity of the diagram 3.3. Now, if λ ∈ Bip(n), then we denote by χ λ the irreducible character of W n associated to λ via Clifford theory (see [9]). The link between the two parametrizations (the ξ's and the χ's) is given by the following result: ...
We construct a subalgebra of dimension of the group algebra of a Weyl group of type containing its Solomon descent's algebra but also the Solomon's descent algebra of the symmetric group. This lead us to a construction of the irreducible characters of the hyperoctahedral groups by using a generalized plactic equivalence.
... (4) Type D n : G = SO(2n) or G = P SO(2n) (adjoint); also Semispin(8) ≃ SO (8). ...
... G = SO(n) and P SO(n). If n = 2, so G is of type D 4 , the three elements x, y, z of Spin(8) are related by automorphisms of Spin (8). Since Spin(8)/ z ≃ SO(8), we conclude Semispin(8) ≃ SO(8), and W lifts by the previous discussion. ...
... The only other exceptional case is D 4 . It follows from a tedious and not very enlightening argument that W lifts to SO (8), uniquely up to T -conjugacy and multiplication by −I, and the lifting to P SO(8) is unique up to T -conjugacy. We leave the details to the reader. ...
Suppose G is a reductive algebraic group, T is a Cartan subgroup, , and W=N/T is the Weyl group. If has order d, it is natural to ask about the orders lifts of w to N. It is straightforward to see that the minimal order of a lift of w has order d or 2d, but it can be a subtle question which holds. We first consider the question of when W itself lifts to a subgroup of N (in which case every element of W lifts to an element of N of the same order). We then consider two natural classes of elements: regular and elliptic. In the latter case all lifts of w are conjugate, and therefore have the same order. We also consider the twisted case.
... Proof. If = ( (0) , (1) ) ∈ 2 ( ), then ⊗ sgn = (( (1) ) * ,( (0) ) * ) by [23,Theorem 5.5.6(c)]. Hence, if ∈ ℱ deg ; , then ⊗ sgn ∈ ℱ deg − ; . ...
... for ∈ 2 ( ) and ∈ ( − ). We will show that if ( , ) ∈ ℱ deg ; × Irr S − , then (23) j ×S − = ( , (1) ) . ...
... Let be an irreducible representation of a parabolic subgroup ′ of . The explicit formula given for the Schur element s , see[23, Theorem 10.5.2] and[33, Lemma 22.12], shows that there exist finitely many integers 1 , 2 , . . ., 1 , 2 , . . ...
The goal of this paper is to compute the cuspidal Calogero-Moser families for all infinite families of finite Coxeter groups, at all parameters. We do this by first computing the symplectic leaves of the associated Calogero-Moser space and then by classifying certain "rigid" modules. Numerical evidence suggests that there is a very close relationship between Calogero-Moser families and Lusztig families. Our classification shows that, additionally, the cuspidal Calogero-Moser families equal cuspidal Lusztig families for the infinite families of Coxeter groups.
... Proposition 2.7.1. (Mackey Formula for Hecke Algebras, [GP,Proposition 9 ...
... Given a finite list s = (s 1 , ..., s n ) of simple reflections, let e s denote the product e s 1 · · · e sn . By Matsumoto's Theorem (see [GP,Theorem 1.2.2]), for any w ∈ I ref the element e s does not depend on the choice of reduced expression w = s 1 . . . s n , and we denote as usual the element e s by e w in this case. ...
... Remark 4.4.2. Our definition of content differs slightly from the definition of content appearing in [GP,Section 10.1.4] because we choose a different convention for the quadratic relations satisfied by the generators T i of H p,q (B n ), i.e. that the quadratic relations should be divisible by (T − 1) rather than (T + 1). ...
For a complex reflection group W with reflection representation , we define and study a natural filtration by Serre subcategories of the category of representations of the rational Cherednik algebra . This filtration refines the filtration by supports and is analogous to the Harish-Chandra series appearing in the representation theory of finite groups of Lie type. Using the monodromy of the Bezrukavnikov-Etingof parabolic restriction functors, we show that the subquotients of this filtration are equivalent to categories of finite-dimensional representations over generalized Hecke algebras. When W is a finite Coxeter group, we give a method for producing explicit presentations of these generalized Hecke algebras in terms of finite-type Iwahori-Hecke algebras. This yields a method for counting the number of irreducible objects in of given support. We apply these techniques to count the number of irreducible representations in of given support for all exceptional Coxeter groups W and all parameters c, including the unequal parameter case. This completes the classification of the finite-dimensional irreducible representations of for exceptional Coxeter groups W in many new cases.
... In Sections 2 and 3 we recall some classical results for Weyl groups of types B n and D n , many of which can be found in Chapters 1 and 5 of [8]. ...
... By [8,Corollary 5.6.4], the field Q is a splitting field for D n . Then all irreducible characters of D n are real valued, and hence all simple CD n -modules are self-dual. ...
We consider the group algebra over the field of complex numbers of the Weyl group of type B (the hyperoctahedral group, or the group of signed permutations) and of the Weyl group of type D (the demihyperoctahedral group, or the group of even-signed permutations), viewed as Lie algebras via the commutator bracket, and determine the structure of the Lie subalgebras generated by the sets of reflections.
... For more information about symmetric algebra, we refer to [8]. (ii) There is a finite set W and a Q-basis {b w | w ∈ W } of H indexed by W such that there is a bijective map ι : ...
... Proposition 4.1. (see[8, Theorem 8.4.6]) The algebra H q has the Schur basis {T w | w ∈ S n } with relations ...
Let n be a positive integer and q be a power of an odd prime. We provide explicit formulas for calculating the orthogonal determinants , where is an orthogonal character of even degree. Moreover, we show that is "odd". This confirms a special case of a conjecture by Richard Parker.
... The first equation in part (b) follows from the braid relations (2), (3) and (12) and the second follows from relations (4), (7) and (13). ...
... It now remains to define φ on B 1 . For this, we draw on the following standard result from Coxeter group theory, proved in [12,Proposition 3.2 ...
----- Please see the pdf file for the actual abstract and important remarks, which could not be put here due to the arXiv length restrictions. ----- This thesis presents a study of the cyclotomic BMW (Birman-Murakami-Wenzl) algebras, introduced by Haring-Oldenburg as a generalization of the BMW algebras associated with the cyclotomic Hecke algebras of type G(k,1,n) (also known as Ariki-Koike algebras) and type B knot theory involving affine/cylindrical tangles. They are shown to be free of rank k^n (2n-1)!! and to have a topological realization as a certain cylindrical analogue of the Kauffman Tangle algebra. Furthermore, the cyclotomic BMW algebras are proven to be cellular, in the sense of Graham and Lehrer. This Ph.D. thesis, completed at the University of Sydney, was submitted September 2007 and passed December 2007.
... forms a C[q, q −1 ]-basis for H 0 (R) (e.g. [8]). The specialization of the algebra H 0 (R) at q = ±1 is the group algebra CW 0 (R). ...
... ) R = E n (n =6,7,8) and v = 3 in the standard labelling. The new labelling of the vertices of the Dynkin diagram is (with l = n − 2):We show an example of a standard tableau of type (R, ...
We introduce and study some affine Hecke algebras of type ADE, generalising the affine Hecke algebras of GL. We construct irreducible calibrated representations and describe the calibrated spectrum. This is done in terms of new families of combinatorial objects equipped with actions of the corresponding Weyl groups. These objects are built from and generalise the usual standard Young tableaux, and are controlled by the considered affine Hecke algebras. By restriction and limiting procedure, we obtain several combinatorial models for representations of finite Hecke algebras and Weyl groups of type ADE. Representations are constructed by explicit formulas, in a seminormal form.
... so that to prove (5) it is enough to show that ...
... ⊙ ρ ι f 0 ,ι f 1 ) and Equation (11) is achieved. Now the result follows from Theorem 4.7 and the following result which is the analogue in type B of the well-known Pieri rule (see [5,Lemma 6.1.3]). ...
In [F. Caselli, Involutory reflection groups and their models, J. Algebra 24 (2010), 370--393] it is constructed a uniform Gelfand model for all non-exceptional irreducible complex reflection groups which are involutory. This model can be naturally decomposed into the direct sum of submodules indexed by symmetric conjugacy classes, and in this paper we present a simple combinatorial description of the irreducible decomposition of these submodules if the group is the wreath product of a cyclic group with a symmetric group. This is attained by showing that such decomposition is compatible with the generalized Robinson-Schensted correspondence for these groups.
... We similarly treat the remaining (exceptional) types of finite irreducible Coxeter groups. Given the formulas of Section 2, this comes to a finite calculation with the character tables of [GP00]. ...
... For completeness, we give N ω for each exceptional group. This is easily done by the propositions in Section 2.4, together with the character tables for these groups, which may be found in [GP00]. ...
In [APS], the authors characterize the partitions of n whose corresponding representations of have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups W. Namely, given a nontrivial multiplicative character of W, we give a closed formula for the number of irreducible representations of W with determinant . For Coxeter groups of type and , this is accomplished by characterizing the bipartitions associated to such representations.
... This proves one part of (1), and the remaining part is a consequence of Tits' deformation theorem (see e.g. [23], §7.4) and of the fact that C W (u) is a free module of finite rank over k[u], by theorem (2) is the consequence of the proposition above together with the fact that the stabilizers of the action of W on W are exactly the normalizers of reflection subgroups. ...
... Therefore, Tits' deformation theorem (see e.g. [23], §7.4) and proposition 5.1 imply the following, where K W denotes a field containing R W . ...
We attach to every Coxeter system (W,S) an extension C_W of the corresponding Iwahori-Hecke algebra. We construct a 1-parameter family of (generically surjective) morphisms from the group algebra of the corresponding Artin group onto C_W. When W is finite, we prove that this algebra is a free module of finite rank which is generically semisimple. When W is the Weyl group of a Chevalley group, C_W naturally maps to the associated Yokonuma-Hecke algebra. When W = S_n this algebra can be identified with a diagram algebra called the algebra of `braids and ties'. The image of the usual braid group in this case is investigated. Finally, we generalize our construction to finite complex reflection groups, thus extending the Broue-Malle-Rouquier construction of a generalized Hecke algebra attached to these groups.
... is of type A, and f ψ is a power of 2, otherwise (see [14,Corollary 9.3.6] and [20, (4.1.1),(4.14. ...
... If we identify V with F n using the basis (12), we view β and Q as forms on F n defined by the formulae (13) respectively (14). LetĜ denote the conformal group of V with respect to the form β or Q, and put G :=Ĝ • . ...
This work completes the classification of the imprimitive irreducible modules, over algebraically closed fields of characteristic 0, of the finite quasisimple groups.
... The explicit expressions for the unit quaternions q 1 , q 2 , . . . q 8 that define the representatives of the conjugacy classes of W (F 4 ) are [85]: The characters of W (F 4 ) were computed in [72] (see also [74,86]). We present the characters of W (F 4 ) for the proper rotations in the Table 14 and for the improper rotations in the Table 15. ...
The Yang-Lee universality class arises when imaginary magnetic field is tuned to its critical value in the paramagnetic phase of the Ising model. In d=2, this non-unitary Conformal Field Theory (CFT) is exactly solvable via the M(2,5) minimal model. As found long ago by von Gehlen using Exact Diagonalizations, the corresponding real-time, quantum critical behavior arises in the periodic Ising spin chain when the imaginary longitudinal magnetic field is tuned to its critical value from below. Even though the Hamiltonian is not Hermitian, the energy levels are real due to the PT symmetry. In this paper, we explore the analogous quantum critical behavior in higher dimensional non-Hermitian Hamiltonians on regularized spheres . For d=3, we use the recently invented, powerful fuzzy sphere method, as well as discretization by the platonic solids cube, icosahedron and dodecaherdron. The low-lying energy levels and structure constants we find are in agreement with expectations from the conformal symmetry. The energy levels are in good quantitative agreement with the high-temperature expansions and with Pad\'e extrapolations of the expansions in Fisher's Euclidean field theory for the Yang-Lee criticality. In the course of this work, we clarify some aspects of matching between operators in this field theory and quasiprimary fields in the M(2,5) minimal model. For d=4, we obtain new results by replacing the with the self-dual polytope called the 24-cell.
... [Al87,Cl06,Ma14], and the Springer theory, e.g. [LA82,GP00,AA08]. The W -module structure of discrete series is indeed expected to fit in the framework of Springer theory and to be computable from some version of Lusztig-Shoji algorithm for complex reflection groups, see [KR02] and [AA08]. ...
This article confirms the prediction that the set of discrete series central character for the graded (affine) Hecke algebra of type coincides with the set of the Heckman-Opdam central characters. Combining with previous cases of Kazhdan-Lusztig, Kriloff, Kriloff-Ram, Opdam-Solleveld, Ciubotaru-Opdam, this completes the classification of discrete series for all the graded Hecke algebras of positive parameters. Main tools include construction of calibrated modules and construction of certain minimally induced modules for discrete series. We also study the anti-sphericiity and Ext-branching laws for some discrete series.
... The following is a construction of irreducible representations of symmetric group S n , see [GP,Chapter 5.4] for a proof. ...
Let G be a general linear group over \BR, \BC, or \BH, or a real unitary group. In this paper, we precisely describe the number of isomorphism classes of irreducible Casselman-Wallach representations of G with a given infinitesimal character and a given associated variety, expressed in terms of certain combinatorial data called painted Young diagrams and assigned Young diagrams.
... V, Section 2, Exercise 3]). For Weyl groups, the exterior powers of the standard reflection representation are well studied (see, for example, [2,4,5,8]). ...
Let W be a group endowed with a finite set S of generators. A representation of W is called a reflection representation of (W,S) if is a (generalized) reflection on V for each generator . In this article, we prove that for any irreducible reflection representation V , all the exterior powers , , are irreducible W -modules, and they are non-isomorphic to each other. This extends a theorem of R. Steinberg which is stated for Euclidean reflection groups. Moreover, we prove that the exterior powers (except for the 0th and the highest power) of two non-isomorphic reflection representations always give non-isomorphic W -modules. This allows us to construct numerous pairwise non-isomorphic irreducible representations for such groups, especially for Coxeter groups.
... The cocenter of the algebra A is the R-module Tr(A) := A/[A, A], where [A, A] is the commutator submodule of A spanned by all commutators of the form [x, y] := xy − yx for x, y ∈ A.The following result was given in[17, Sec. 7.1.7]. ...
In this paper, we obtain bases for the center and cocenter of the q-Schur superalgebra S(1|1,r), in terms of its relative norm basis and normalized basis.
... It is known that all but a finite number of complex reflection groups satisfy this hypothesis, see [AK], [BM] and [Ari], and it is conjectured to hold for all complex reflection groups. By Tits' deformation theorem (see, for example, [GP,Theorem 7 ...
Let W be a complex reflection group. We formulate a conjecture relating blocks of the corresponding restricted rational Cherednik algebras and Rouquier families for cyclotomic Hecke algebras. We verify the conjecture in the case that W is a wreath product of a symmetric group with a cyclic group of order l.
... The finite Hecke algebra H n is a symmetric algebra, see for example [GP00]. This implies that the Chern character map h n : K 0 (H n −fmod) −→ Tr(H n −fmod) is an isomorphism. ...
We give an explicit description of the trace, or Hochschild homology, of the quantum Heisenberg category defined by Licata and Savage. We also show that as an algebra, it is isomorphic to "half" of a central extension of the elliptic Hall algebra of Burban and Schiffmann, specialized at . A key step in the proof may be of independent interest: we show that the sum (over n) of the Hochschild homologies of the positive affine Hecke algebras is again an algebra, and that this algebra injects into both the elliptic Hall algebra and the trace of the q-Heisenberg category. Finally, we show that a natural action of the trace algebra on the space of symmetric functions agrees with the specialization of an action constructed by Schiffmann and Vasserot using Hilbert schemes.
... The irreducible complex representations of B n are in natural bijection with double partitions of n, that is, ordered pairs of partitions (λ, µ) such that |λ| + |µ| = n. These irreducible representations are constructed from representations of the symmetric group S n ; this construction is described (for example) in Geck-Pfeiffer [GP00]. From the canonical surjection B n → S n we can pull back representations of S n to B n . ...
A result of Lehrer describes a beautiful relationship between topological and combinatorial data on certain families of varieties with actions of finite reflection groups. His formula relates the cohomology of complex varieties to point counts on associated varieties over finite fields. Church, Ellenberg, and Farb use their representation stability results on the cohomology of flag manifolds, together with classical results on the cohomology rings, to prove asymptotic stability for "polynomial" statistics on associated varieties over finite fields. In this paper we investigate the underlying algebraic structure of these families' cohomology rings that makes the formulas convergent. We prove that asymptotic stability holds in general for subquotients of FI-algebras finitely generated in degree at most one, a result that is in a sense sharp. As a consequence, we obtain convergence results for polynomial statistics on the set of maximal tori in and that are invariant under the Frobenius morphism. Our results also give a new proof of the stability theorem for invariant maximal tori in due to Church-Ellenberg-Farb.
... Proof. The tables in [44,Appendix F] show that the Brauer tree of the principal block of the Hecke algebra End OG (R G T (O)) is a line with r + 1 vertices, with leaves corresponding to the trivial and sign characters. So, T has a full subgraph L that is a line with r + 1 vertices and with leaves 1 and St. ...
In this paper we complete the determination of the Brauer trees of unipotent blocks (with cyclic defect groups) of finite groups of Lie type. These trees were conjectured by the first author. As a consequence, the Brauer trees of principal -blocks of finite groups are known for .
... Given V a representation of G n,d , let the χ V be the associated character to it and let V be the corresponding local system on Conf 0 n . Initially this construction is done over C but since since every irreducible representation of G n,d can be defined over Z[µ d ] (see [12], [8]), the local system V determines an l-adic sheaf and we shall not make a distinction between the two objects. ...
In this paper we prove an explicit version of a function field analogue of a classical result of Odoni about norms in number fields in the case of a cyclic Galois extensions. In the particular case of a quadratic extension, we recover the result of Bary-Soroker, Smilanski, and Wolf which deals with finding asymptotics for a function field version on sums of two squares, improved upon by Gorodetsky , and reproved by the author in his Ph.D thesis using the method of this paper. The main tool is a twisted Grothendieck Lefschetz trace formula, inspired by the work of Church, Farb and Ellenberg on representation stability and asymptotic for point counts on varieties. Using a combinatorial description of the cohomology we obtain a precise quantitative result which works in the regime, and a new type of homological stability phenomena, which arises from the computation of certain inner products of representations.
... Let c ∈ C[S] adG and set q = exp(2π √ −1c) ∈ C * [S] adG . Then O c is semisimple if and only if H q (G) is semisimple.If G is a Weyl group then there a good conditions which ensure the semisimplicity of H q (G), see for instance[64, Chapter 9] and[3, Theorem 3.29]. Conversely, for arbitrary G the corollary combined with(13) shows that if if the subgroup of C * generated by the q(s) is torsion-free then H q (G) is semisimple,[126, Theorem 3.4]. ...
We survey recent results on the representation theory of symplectic reflection algebras, focusing particularly on connections with symplectic quotient singularities and their resolutions, spaces of representations of quivers, and on category O.
... For λ ⊢ n, let S λ be the corresponding irreducible representation of the symmetric group S n . If we fix a basis for any H n (q)-irreducible W λ , it is well known (see, e.g., [7,Thm. 8.1.7]) ...
We construct an action of the Hecke algebra on a quotient of the polynomial ring , where . The dimension of our quotient ring is the number of k-block ordered set partitions of . This gives a quantum analog of a construction of Haglund-Rhoades-Shimozono and interpolates between their result at and work of Huang-Rhoades at .
... , X n pairwise commute ( [ArKo,Lemma 3.3.(2)]). Moreover, Matsumoto's theorem (see, for instance, [GePf,Theorem 1.2.2]) ensures that (1.2e) and (1.2f) allow us to define T w := T a1 · · · T am for any reduced expression w = s a1 · · · s am ∈ S n , where s a ∈ S n is the transposition (a, a + 1). ...
Given a quiver automorphism with nice properties, we give a presentation of the fixed subalgebra of the associated cyclotomic quiver Hecke algebra. Generalising an isomorphism of Brundan and Kleshchev between the cyclotomic Hecke algebra of type G(r,1,n) and the cyclotomic quiver Hecke algebra of type A, we apply the previous result to find a presentation of the cyclotomic Hecke algebra of type G(r,p,n) which looks very similar to the one of a cyclotomic quiver Hecke algebra. In the meanwhile, we give an explicit isomorphism which realises a well-known Morita equivalence between Ariki-Koike algebras.
... This follows from [18, 3.2] and [5, 2.13.1] (see also [10,Exercise 5.6]). Then, for general x and y, Corollary 3.6 is equivalent to: ...
We compute two-sided cells of Weyl groups of type B for the "asymptotic" choice of parameters. We also obtain some partial results concerning Kazhdan-Lusztig conjectures in this particular case.
... This follows from the Gaschütz-Ikeda Lemma (cf. [8,Lemma 7.1.11]), which is a special case of Higman's criterion for modules over symmetric algebras in Broué [4]. ...
We identify a class of symmetric algebras over a complete discrete valuation ring of characteristic zero to which the characterisation of Kn\"orr lattices in terms of stable endomorphism rings in the case of finite group algebras, can be extended. This class includes finite group algebras, their blocks and source algebras and Hopf orders. We also show that certain arithmetic properties of finite group representations extend to this class of algebras. Our results are based on an explicit description of Tate duality for lattices over symmetric -algebras whose extension to the quotient field of is separable.
... where 0 = c λ ∈ R; see [20,Chapter 7]. We have ...
More than 10 years ago, Dipper, James and Murphy developped the theory of Specht modules for Hecke algebras of type . More recently, using Lusztig's a-function, Geck and Rouquier showed how to obtain parametrisations of the irreducible representations of Hecke algebras (of any finite type) in terms of so-called canonical basic sets. For certain values of the parameters in type , combinatorial descriptions of these basic sets were found by Jacon, based on work of Ariki and Foda-Leclerc-Okado-Thibon-Welsh. Here, we consider the canonical basic sets for all the remaining choices of the parameters.
... Each element w ∈ W can have several different reduced expressions that represent it. The following theorem, called Matsumoto's Theorem [4], indicates how all of the reduced expressions for a given group element are related. Proposition 2.1. ...
The Temperley--Lieb algebra is a finite dimensional associative algebra that arose in the context of statistical mechanics and occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type A. It is often realized in terms of a certain diagram algebra, where every diagram can be written as a product of "simple diagrams." These factorizations correspond precisely to factorizations of the so-called fully commutative elements of the Coxeter group that index a particular basis. Given a reduced factorization of a fully commutative element, it is straightforward to construct the corresponding diagram. On the other hand, it is generally difficult to reconstruct the factorization given an arbitrary diagram. We present an efficient algorithm for obtaining a reduced factorization for a given diagram.
We study the quasi-admissibility and raisability of some nilpotent orbits of a covering group. In particular, we determine the degree of the cover such that a given split nilpotent orbit is quasi-admissible and non-raisable. The speculated wavefront sets of theta representations are also computed explicitly and are shown to be quasi-admissible and non-raisable. Lastly, we determine the leading coefficients in the Harish-Chandra character expansion of the theta representations of covers of the general linear groups.
An automorphism of a building is called uniclass if the Weyl distance between any chamber and its image lies in a single (twisted) conjugacy class of the Coxeter group. In this paper we characterise uniclass automorphisms of spherical buildings in terms of their fixed structure. For this purpose we introduce the notion of a Weyl substructure in a spherical building. We also link uniclass automorphisms to the Freudenthal–Tits magic square.
The exceptional complex reflection groups of rank 2 are partitioned into three families. We construct explicit matrix models for the Hecke algebras associated to the maximal groups in the tetrahedral and octahedral family, and use them to verify the BMM symmetrising trace conjecture for all groups in these two families, providing evidence that a similar strategy might apply for the icosahedral family.
We complete the determination of the ℓ‐block distribution of characters for quasi‐simple exceptional groups of Lie type up to some minor ambiguities relating to non‐uniqueness of Jordan decomposition. For this, we first determine the ℓ‐block distribution for finite reductive groups whose ambient algebraic group defined in characteristic different from ℓ has connected centre. As a consequence, we derive a compatibility between ℓ‐blocks, ee‐Harish‐Chandra series and Jordan decomposition. Further, we apply our results to complete the proof of Robinson's conjecture on defects of characters.
This paper uses Lusztig varieties to give central elements of the Iwahori-Hecke algebra corresponding to unipotent conjugacy classes in the finite Chevalley group . We explain how these central elements are related to Macdonald polynomials and how this provides a framework for generalizing integral form and modified Macdonald polynomials to Lie types other than . The key steps are to recognize (a) that counting points in Lusztig varieties is equivalent to computing traces on the Hecke algebras, (b) that traces on the Hecke algebra determine elements of the center of the Hecke algebra, (c) that the Geck-Rouquier basis elements of the center of the Hecke algebra produce an ‘expansion matrix’, (d) that the parabolic subalgebras of the Hecke algebra produce a ‘contraction matrix’ and (e) that the combination ‘expansion-contraction’ is the plethystic transformation that relates integral form Macdonald polynomials and modified Macdonald polynomials.
We provide an algorithmic framework for the computation of explicit representing matrices for all irreducible representations of a generalized symmetric group \Grin_n, i.e., a wreath product of cyclic group of order r with the symmetric group \Symm_n. The basic building block for this framework is the Specht matrix, a matrix with entries 0 and , defined in terms of pairs of certain words. Combinatorial objects like Young diagrams and Young tableaus arise naturally from this setup. In the case , we recover Young's natural representations of the symmetric group. For general r, a suitable notion of pairs of r-words is used to extend the construction to generalized symmetric groups. Separately, for , where \Grin_n is the Weyl group of type , a different construction is based on a notion of pairs of biwords.
We study the Howe correspondence for unipotent representations of irreducible dual pairs and , where denotes the finite field with q elements (q odd) and . We show how to extract extremal (i.e. minimal and maximal) irreducible subrepresentations from the image of under the correpondence of a unipotent representation of G.
From the combinatorial characterizations of the right, left, and two-sided Kazhdan-Lusztig cells of the symmetric group, 'RSK bases' are constructed for certain quotients by two-sided ideals of the group ring and the Hecke algebra. Applications to invariant theory, over various base rings, of the general linear group and representation theory of the symmetric group are discussed.
We prove that two Springer maps over a nilpotent orbit closure with the same degree are connected by stratified Mukai flops and the latter is obtained by extremal contractions of a natural resolution of the nilpotent orbit closure.
We describe completely the link invariants constructed using Markov traces on the Yokonuma-Hecke algebras in terms of the linking matrix and the HOMFLYPT polynomials of sublinks.
In this paper we consider the (affine) Schur algebra introduced by Vign\'eras as the endomorphism algebra of certain permutation modules for the Iwahori-Matsumoto Hecke algebra. This algebra describes, for a general linear group over a p-adic field, a large part of the unipotent block over fields of characteristic different from p. We show that this Schur algebra is, after a suitable completion, isomorphic to the quiver Schur algebra attached to the cyclic quiver. The isomorphism is explicit, but nontrivial. As a consequence, the completed (affine) Schur algebra inherits a grading. As a byproduct we obtain a detailed description of the algebra with a basis adapted to the geometric basis of quiver Schur algebras. We illustrate the grading in the explicit example of in characteristic $3
We prove that the Drinfeld halfspace is essentially the only Deligne-Lusztig variety which is at the same time a period domain over a finite field. This is done by comparing a cohomology vanishing theorem for DL-varieties, due to Digne, Michel, and Rouquier, with a non-vanishing theorem for PD, due to the first author. We also discuss an affineness criterion for DL-varieties.
We give explicit descriptions of the adjoint group of the Coxeter quandle associated with an arbitrary Coxeter group W. The adjoint group of turns out to be an intermediate group between W and the corresponding Artin group , and fits into a central extension of W by a finitely generated free abelian group. We construct 2-cocycles of W corresponding to the central extension. In addition, we prove that the commutator subgroup of the adjoint group of is isomorphic to the commutator subgroup of W. Finally, the root system associated with a Coxeter group W turns out to be a rack. We prove that the adjoint group of is isomorphic to the adjoint group of .
We extend the classification of finite Weyl groupoids of rank two. Then we generalize these Weyl groupoids to `reflection groupoids' by admitting non-integral entries of the Cartan matrices. This leads to the unexpected observation that the spectrum of the cluster algebra of type completely describes the set of finite reflection groupoids of rank two with 2n objects.
This paper studies affine Deligne-Lusztig varieties in the affine flag manifold of a split group. Among other things, it proves emptiness for certain of these varieties, relates some of them to those for Levi subgroups, extends previous conjectures concerning their dimensions, and generalizes the superset method.
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