0-Hecke algebras

1979 ◽  
Vol 27 (3) ◽  
pp. 337-357 ◽  
Author(s):  
P. N. Norton
Keyword(s):  
Coxeter Group ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Irreducible Representations ◽  
One Dimensional ◽  
Direct Sum ◽  
Natural Way

AbstractThe structure of a 0-Hecke algebra H of type (W, R) over a field is examined. H has 2n distinct irreducible representations, where n = ∣R∣, all of which are one-dimensional, and correspond in a natural way with subsets of R. H can be written as a direct sum of 2n indecomposable left ideals, in a similar way to Solomon's (1968) decomposition of the underlying Coxeter group W.

scholarly journals Parameters for generalized Hecke algebras in type B

2019 ◽  
Vol 18 (09) ◽  
pp. 1950173
Author(s):  
Max Murin ◽  
Seth Shelley-Abrahamson
Keyword(s):  
Recent Work ◽  
Parabolic Subgroup ◽  
Coxeter Group ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Irreducible Representations ◽  
Type B ◽  
Full Support

The irreducible representations of full support in the rational Cherednik category [Formula: see text] attached to a Coxeter group [Formula: see text] are in bijection with the irreducible representations of an associated Iwahori–Hecke algebra. Recent work has shown that the irreducible representations in [Formula: see text] of arbitrary given support are similarly governed by certain generalized Hecke algebras. In this paper, we compute the parameters for these generalized Hecke algebras in the remaining previously unknown cases, corresponding to the parabolic subgroup [Formula: see text] in [Formula: see text] for [Formula: see text] and [Formula: see text].

scholarly journals A quantum probabilistic approach to Hecke algebras for 𝔭-adic PGL2

2018 ◽  
Vol 21 (03) ◽  
pp. 1850015
Author(s):  
Takehiro Hasegawa ◽  
Hayato Saigo ◽  
Seiken Saito ◽  
Shingo Sugiyama
Keyword(s):  
Inversion Formula ◽  
Probabilistic Approach ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Quantum Probability ◽  
Probabilistic Interpretation ◽  
Fourier Inversion ◽  
Fourier Inversion Formula ◽  
The Subject

The subject of the present paper is an application of quantum probability to [Formula: see text]-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for [Formula: see text], where [Formula: see text] is a [Formula: see text]-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for [Formula: see text].

scholarly journals Positive energy representations of Sobolev diffeomorphism groups of the circle

2020 ◽  
Vol 11 (1) ◽  
Author(s):  
Sebastiano Carpi ◽  
Simone Del Vecchio ◽  
Stefano Iovieno ◽  
Yoh Tanimoto
Keyword(s):  
Unitary Representation ◽  
Von Neumann Algebras ◽  
Universal Covering ◽  
Irreducible Representations ◽  
Positive Energy ◽  
Diffeomorphism Groups ◽  
Von Neumann ◽  
Direct Sum ◽  
Covering Groups

AbstractWe show that any positive energy projective unitary representation of $$\mathrm{Diff}_+(S^1)$$ Diff + ( S 1 ) extends to a strongly continuous projective unitary representation of the fractional Sobolev diffeomorphisms $$\mathcal {D}^s(S^1)$$ D s ( S 1 ) for any real $$s>3$$ s > 3 , and in particular to $$C^k$$ C k -diffeomorphisms $$\mathrm{Diff}_+^k(S^1)$$ Diff + k ( S 1 ) with $$k\ge 4$$ k ≥ 4 . A similar result holds for the universal covering groups provided that the representation is assumed to be a direct sum of irreducibles. As an application we show that a conformal net of von Neumann algebras on $$S^1$$ S 1 is covariant with respect to $$\mathcal {D}^s(S^1)$$ D s ( S 1 ) , $$s > 3$$ s > 3 . Moreover every direct sum of irreducible representations of a conformal net is also $$\mathcal {D}^s(S^1)$$ D s ( S 1 ) -covariant.

Note on Weight Spaces of Irreducible Linear Representations

1968 ◽  
Vol 11 (3) ◽  
pp. 399-403 ◽  
Author(s):  
F. W. Lemire
Keyword(s):  
Irreducible Representation ◽  
Weight Function ◽  
Wide Class ◽  
Irreducible Representations ◽  
Weight Space ◽  
Closed Field ◽  
One Dimensional ◽  
Highest Weight ◽  
Linear Representations ◽  
Finite Dimensional

Let L denote a finite dimensional, simple Lie algebra over an algebraically closed field F of characteristic zero. It is well known that every weight space of an irreducible representation (ρ, V) admitting a highest weight function is finite dimensional. In a previous paper [2], we have established the existence of a wide class of irreducible representations which admit a one-dimensional weight space but no highest weight function. In this paper we show that the weight spaces of all such representations are finite dimensional.

scholarly journals Mixed Perverse Sheaves on Flag Varieties for Coxeter Groups

2019 ◽  
Vol 72 (1) ◽  
pp. 1-55
Author(s):  
Pramod N. Achar ◽  
Simon Riche ◽  
Cristian Vay
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Grothendieck Group ◽  
General Setting ◽  
Hecke Algebra ◽  
Abelian Category ◽  
Flag Varieties ◽  
Perverse Sheaves

AbstractIn this paper we construct an abelian category of mixed perverse sheaves attached to any realization of a Coxeter group, in terms of the associated Elias–Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias–Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.

scholarly journals A decomposition theorem for homogeneous algebras

2002 ◽  
Vol 72 (1) ◽  
pp. 47-56 ◽  
Author(s):  
L. G. Sweet ◽  
J. A. Macdougall
Keyword(s):  
Nonzero Element ◽  
Decomposition Theorem ◽  
Small Dimension ◽  
Infinite Field ◽  
Left Multiplication ◽  
One Dimensional ◽  
Direct Sum ◽  
Homogeneous Algebras ◽  
The One ◽  
Homogeneous Algebra

AbstractAn algebra A is homogeneous if the automorphism group of A acts transitively on the one dimensional subspaces of A. Suppose A is a homogeneous algebra over an infinite field k. Let La denote left multiplication by any nonzero element a ∈ A. Several results are proved concerning the structure of A in terms of La. In particular, it is shown that A decomposes as the direct sum A = ker La Im La. These results are then successfully applied to the problem of classifying the infinite homogeneous algebras of small dimension.

scholarly journals Leading coefficients and cellular bases of Hecke algebras

2009 ◽  
Vol 52 (3) ◽  
pp. 653-677 ◽  
Author(s):  
Meinolf Geck
Keyword(s):  
Weyl Group ◽  
Coxeter Group ◽  
Cellular Structure ◽  
Hecke Algebras ◽  
Point Of View ◽  
Reflection Groups ◽  
Cellular Basis ◽  
Complex Reflection Groups ◽  
Finite Coxeter Group ◽  
New Construction

AbstractLet H be the generic Iwahori–Hecke algebra associated with a finite Coxeter group W. Recently, we have shown that H admits a natural cellular basis in the sense of Graham and Lehrer, provided that W is a Weyl group and all parameters of H are equal. The construction involves some data arising from the Kazhdan–Lusztig basis {Cw} of H and Lusztig's asymptotic ring J}. We attempt to study J and its representation theory from a new point of view. We show that J can be obtained in an entirely different fashion from the generic representations of H, without any reference to {Cw}. We then extend the construction of the cellular basis to the case where W is not crystallographic. Furthermore, if H is a multi-parameter algebra, we see that there always exists at least one cellular structure on H. Finally, the new construction of J may be extended to Hecke algebras associated with complex reflection groups.

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