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Journal articles on the topic 'Hecke algebra'

Author: Grafiati

Published: 4 June 2021

Last updated: 20 February 2023

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1

Aubert, Anne-Marie, Paul Baum, Roger Plymen, and Maarten Solleveld. "HECKE ALGEBRAS FOR INNER FORMS OF -ADIC SPECIAL LINEAR GROUPS." Journal of the Institute of Mathematics of Jussieu 16, no. 2 (May 5, 2015): 351–419. http://dx.doi.org/10.1017/s1474748015000079.

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Let$F$be a non-Archimedean local field, and let$G^{\sharp }$be the group of$F$-rational points of an inner form of$\text{SL}_{n}$. We study Hecke algebras for all Bernstein components of$G^{\sharp }$, via restriction from an inner form$G$of$\text{GL}_{n}(F)$.For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth$G^{\sharp }$-representations. This algebra comes from an idempotent in the full Hecke algebra of$G^{\sharp }$, and the idempotent is derived from a type for$G$. We show that the Hecke algebras for Bernstein components of$G^{\sharp }$are similar to affine Hecke algebras of type$A$, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

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2

Lee, Kyu-Hwan. "Iwahori-Hecke Algebras of SL2 over 2-Dimensional Local Fields." Canadian Journal of Mathematics 62, no. 6 (December 14, 2010): 1310–24. http://dx.doi.org/10.4153/cjm-2010-072-x.

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AbstractIn this paper we construct an analogue of Iwahori–Hecke algebras of SL2 over 2-dimensional local fields. After considering coset decompositions of double cosets of a Iwahori subgroup, we define a convolution product on the space of certain functions on SL2, and prove that the product is well-defined, obtaining a Hecke algebra. Then we investigate the structure of the Hecke algebra. We determine the center of the Hecke algebra and consider Iwahori–Matsumoto type relations.

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3

OGIEVETSKY, O. V., and L. POULAIN D'ANDECY. "ON REPRESENTATIONS OF CYCLOTOMIC HECKE ALGEBRAS." Modern Physics Letters A 26, no. 11 (April 10, 2011): 795–803. http://dx.doi.org/10.1142/s0217732311035377.

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An approach, based on Jucys–Murphy elements, to the representation theory of cyclotomic Hecke algebras is developed. The maximality (in the cyclotomic Hecke algebra) of the set of the Jucys–Murphy elements is established. A basis of the cyclotomic Hecke algebra is suggested; this basis is used to establish the flatness of the deformation without using the representation theory.

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4

Ciubotaru, Dan, Eric M. Opdam, and Peter E. Trapa. "Algebraic and analytic Dirac induction for graded affine Hecke algebras." Journal of the Institute of Mathematics of Jussieu 13, no. 3 (March 13, 2013): 447–86. http://dx.doi.org/10.1017/s147474801300008x.

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AbstractWe define the algebraic Dirac induction map ${\mathrm{Ind} }_{D} $ for graded affine Hecke algebras. The map ${\mathrm{Ind} }_{D} $ is a Hecke algebra analog of the explicit realization of the Baum–Connes assembly map in the $K$-theory of the reduced ${C}^{\ast } $-algebra of a real reductive group using Dirac operators. The definition of ${\mathrm{Ind} }_{D} $ is uniform over the parameter space of the graded affine Hecke algebra. We show that the map ${\mathrm{Ind} }_{D} $ defines an isometric isomorphism from the space of elliptic characters of the Weyl group (relative to its reflection representation) to the space of elliptic characters of the graded affine Hecke algebra. We also study a related analytically defined global elliptic Dirac operator between unitary representations of the graded affine Hecke algebra which are realized in the spaces of sections of vector bundles associated to certain representations of the pin cover of the Weyl group. In this way we realize all irreducible discrete series modules of the Hecke algebra in the kernels (and indices) of such analytic Dirac operators. This can be viewed as a graded affine Hecke algebra analog of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and to Atiyah and Schmid.

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5

Yang, Guiyu, and Yanbo Li. "Standardly based algebras and 0-Hecke algebras." Journal of Algebra and Its Applications 14, no. 10 (September 2015): 1550141. http://dx.doi.org/10.1142/s0219498815501418.

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In this paper we prove that standardly based algebras are invariant under Morita equivalences. As an application, we prove 0-Hecke algebras and 0-Schur algebras are standardly based algebras. From this point of view, we give a new way to construct the simple modules of 0-Hecke algebras, and prove that the dimension of the center of a symmetric 0-Hecke algebra is not less than the number of its simple modules.

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6

SAUTER, JULIA. "FROM COMPLETE TO PARTIAL FLAGS IN GEOMETRIC EXTENSION ALGEBRAS." Glasgow Mathematical Journal 60, no. 1 (March 13, 2017): 111–21. http://dx.doi.org/10.1017/s0017089516000598.

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AbstractA geometric extension algebra is an extension algebra of a semi-simple perverse sheaf (allowing shifts), e.g., a push-forward of the constant sheaf under a projective map. Particular nice situations arise for collapsings of homogeneous vector bundles over homogeneous spaces. In this paper, we study the relationship between partial flag and complete flag cases. Our main result is that the locally finite modules over the geometric extension algebras are related by a recollement. As examples, we investigate parabolic affine nil Hecke algebras, geometric extension algebras associated with parabolic Springer maps and an example of Reineke of a parabolic quiver-graded Hecke algebra.

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7

Braverman, Alexander, and David Kazhdan. "Some Examples of Hecke Algebras for Two-Dimensional Local Fields." Nagoya Mathematical Journal 183 (2006): 57–84. http://dx.doi.org/10.1017/s0027763000009314.

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Let K be a local non-archimedian field, F = K((t)) and let G be a split semi-simple group. The purpose of this paper is to study certain analogs of spherical and Iwahori Hecke algebras for representations of the group G = G(F) and its central extension Ĝ. For instance our spherical Hecke algebra corresponds to the subgroup G (A) ⊂ G(F) where A ⊂ F is the subring OK((t)) where OK ⊂ K is the ring of integers. It turns out that for generic level (cf. [4]) the spherical Hecke algebra is trivial; however, on the critical level it is quite large. On the other hand we expect that the size of the corresponding Iwahori-Hecke algebra does not depend on a choice of a level (details will be considered in another publication).

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8

Orellana, R. C. "Weights of Markov traces on Hecke algebras." Journal für die reine und angewandte Mathematik (Crelles Journal) 1999, no. 508 (March 12, 1999): 157–78. http://dx.doi.org/10.1515/crll.1999.508.157.

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Abstract We compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the Hecke algebra of type Bn and type Dn. In order to prove the weight formula, we define representations of the Hecke algebra of type B onto a reduced Hecke algebra of type A. To compute the weights for type D we use the inclusion of the Hecke algebra of type D into the Hecke algebra of type B.

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9

Barbasch, Dan, and Allen Moy. "Peter–Weyl Iwahori Algebras." Canadian Journal of Mathematics 72, no. 5 (June 21, 2019): 1304–23. http://dx.doi.org/10.4153/s0008414x19000324.

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AbstractThe Peter–Weyl idempotent $e_{\mathscr{P}}$ of a parahoric subgroup $\mathscr{P}$ is the sum of the idempotents of irreducible representations of $\mathscr{P}$ that have a nonzero Iwahori fixed vector. The convolution algebra associated with $e_{\mathscr{P}}$ is called a Peter–Weyl Iwahori algebra. We show that any Peter–Weyl Iwahori algebra is Morita equivalent to the Iwahori–Hecke algebra. Both the Iwahori–Hecke algebra and a Peter–Weyl Iwahori algebra have a natural conjugate linear anti-involution $\star$, and the Morita equivalence preserves irreducible hermitian and unitary modules. Both algebras have another anti-involution, denoted by $\bullet$, and the Morita equivalence preserves irreducible and unitary modules for $\bullet$.

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10

Larsen, N. S., and Iain Raeburn. "Faithful representations of crossed products by actions of $\boldsymbol N^k$." MATHEMATICA SCANDINAVICA 89, no. 2 (December 1, 2001): 283. http://dx.doi.org/10.7146/math.scand.a-14342.

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We study a family of semigroup crossed products arising from actions of $\boldsymbol N^k$ by endomorphisms of groups. These include the Hecke algebra arising in the Bost-Connes analysis of phase transitions in number theory, and other Hecke algebras considered by Brenken. Our main theorem is a characterisation of the faithful representations of these crossed products, and generalises a similar theorem for the Bost-Connes algebra due to Laca and Raeburn.

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11

Opdam, Eric, and Maarten Solleveld. "Homological algebra for affine Hecke algebras." Advances in Mathematics 220, no. 5 (March 2009): 1549–601. http://dx.doi.org/10.1016/j.aim.2008.11.002.

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12

Abe, Noriyuki. "Modulo p parabolic induction of pro-p-Iwahori Hecke algebra." Journal für die reine und angewandte Mathematik (Crelles Journal) 2019, no. 749 (April 1, 2019): 1–64. http://dx.doi.org/10.1515/crelle-2016-0043.

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Abstract We study the structure of parabolic inductions of a pro-p-Iwahori Hecke algebra. In particular, we give a classification of irreducible modulo p representations of pro-p-Iwahori Hecke algebras in terms of supersingular representations. Since supersingular representations are classified by Ollivier and Vignéras, it completes the classification of irreducible modulo p representations.

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13

McGerty, Kevin. "On the centre of the cyclotomic Hecke algebra of G(m, 1, 2)." Proceedings of the Edinburgh Mathematical Society 55, no. 2 (April 12, 2012): 497–506. http://dx.doi.org/10.1017/s0013091510001264.

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AbstractWe compute the centre of the cyclotomic Hecke algebra attached to G(m, 1, 2) and show that if q ≠ 1, it is equal to the image of the centre of the affine Hecke algebra Haff2. We also briefly discuss what is known about the relation between the centre of an arbitrary cyclotomic Hecke algebra and the centre of the affine Hecke algebra of type A.

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14

Goundaroulis, D., and S. Lambropoulou. "Classical link invariants from the framizations of the Iwahori–Hecke algebra and the Temperley–Lieb algebra of type A." Journal of Knot Theory and Its Ramifications 26, no. 09 (August 2017): 1743005. http://dx.doi.org/10.1142/s0218216517430052.

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In this paper, we first present the construction of the new 2-variable classical link invariants arising from the Yokonuma–Hecke algebras [Formula: see text], which are not topologically equivalent to the Homflypt polynomial. We then present the algebra [Formula: see text] which is the appropriate Temperley–Lieb analogue of [Formula: see text], as well as the related 1-variable classical link invariants, which in turn are not topologically equivalent to the Jones polynomial. Finally, we present the algebra of braids and ties which is related to the Yokonuma–Hecke algebra, and also its quotient, the partition Temperley–Lieb algebra [Formula: see text] and we prove an isomorphism of this algebra with a subalgebra of [Formula: see text].

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15

Böckle, Gebhard, Chandrashekhar B. Khare, and Jeffrey Manning. "Wiles defect for Hecke algebras that are not complete intersections." Compositio Mathematica 157, no. 9 (August 16, 2021): 2046–88. http://dx.doi.org/10.1112/s0010437x21007454.

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In his work on modularity theorems, Wiles proved a numerical criterion for a map of rings $R\to T$ to be an isomorphism of complete intersections. He used this to show that certain deformation rings and Hecke algebras associated to a mod $p$ Galois representation at non-minimal level are isomorphic and complete intersections, provided the same is true at minimal level. In this paper we study Hecke algebras acting on cohomology of Shimura curves arising from maximal orders in indefinite quaternion algebras over the rationals localized at a semistable irreducible mod $p$ Galois representation $\bar {\rho }$. If $\bar {\rho }$ is scalar at some primes dividing the discriminant of the quaternion algebra, then the Hecke algebra is still isomorphic to the deformation ring, but is not a complete intersection, or even Gorenstein, so the Wiles numerical criterion cannot apply. We consider a weight-2 newform $f$ which contributes to the cohomology of the Shimura curve and gives rise to an augmentation $\lambda _f$ of the Hecke algebra. We quantify the failure of the Wiles numerical criterion at $\lambda _f$ by computing the associated Wiles defect purely in terms of the local behavior at primes dividing the discriminant of the global Galois representation $\rho _f$ which $f$ gives rise to by the Eichler–Shimura construction. One of the main tools used in the proof is Taylor–Wiles–Kisin patching.

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16

Zhong, Changlong. "ON THE FORMAL AFFINE HECKE ALGEBRA." Journal of the Institute of Mathematics of Jussieu 14, no. 4 (June 9, 2014): 837–55. http://dx.doi.org/10.1017/s1474748014000188.

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We study the action of the formal affine Hecke algebra on the formal group algebra, and show that the the formal affine Hecke algebra has a basis indexed by the Weyl group as a module over the formal group algebra. We also define a concept called the normal formal group law, which we use to simplify the relations of the generators of the formal affine Demazure algebra and the formal affine Hecke algebra.

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17

Chlouveraki, Maria, Jesús Juyumaya, Konstantinos Karvounis, and Sofia Lambropoulou. "Identifying the Invariants for Classical Knots and Links from the Yokonuma–Hecke Algebras." International Mathematics Research Notices 2020, no. 1 (March 15, 2018): 214–86. http://dx.doi.org/10.1093/imrn/rny013.

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Abstract We announce the existence of a family of new 2-variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma–Hecke algebra of type A. Yokonuma–Hecke algebras are generalizations of Iwahori–Hecke algebras, and this family contains the HOMFLYPT polynomial, the famous 2-variable invariant for classical links arising from the Iwahori–Hecke algebra of type A. We show that these invariants are topologically equivalent to the HOMFLYPT polynomial on knots, but not on links, by providing pairs of HOMFLYPT-equivalent links that are distinguished by our invariants. In order to do this, we prove that our invariants can be defined diagrammatically via a special skein relation involving only crossings between different components. We further generalize this family of invariants to a new 3-variable skein link invariant that is stronger than the HOMFLYPT polynomial. Finally, we present a closed formula for this invariant, by W. B. R. Lickorish, that uses HOMFLYPT polynomials of sublinks and linking numbers of a given oriented link.

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18

Bowman, Chris, Anton Cox, Amit Hazi, and Dimitris Michailidis. "Path combinatorics and light leaves for quiver Hecke algebras." Mathematische Zeitschrift 300, no. 3 (September 29, 2021): 2167–203. http://dx.doi.org/10.1007/s00209-021-02829-0.

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AbstractWe recast the classical notion of “standard tableaux" in an alcove-geometric setting and extend these classical ideas to all “reduced paths" in our geometry. This broader path-perspective is essential for implementing the higher categorical ideas of Elias–Williamson in the setting of quiver Hecke algebras. Our first main result is the construction of light leaves bases of quiver Hecke algebras. These bases are richer and encode more structural information than their classical counterparts, even in the case of the symmetric groups. Our second main result provides path-theoretic generators for the “Bott–Samelson truncation" of the quiver Hecke algebra.

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19

Murin, Max, and Seth Shelley-Abrahamson. "Parameters for generalized Hecke algebras in type B." Journal of Algebra and Its Applications 18, no. 09 (July 17, 2019): 1950173. http://dx.doi.org/10.1142/s0219498819501731.

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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].

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20

GREEN, R. M. "GENERALIZED TEMPERLEY–LIEB ALGEBRAS AND DECORATED TANGLES." Journal of Knot Theory and Its Ramifications 07, no. 02 (March 1998): 155–71. http://dx.doi.org/10.1142/s0218216598000103.

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We give presentations, by means of diagrammatic generators and relations, of the analogues of the Temperley–Lieb algebras associated as Hecke algebra quotients to Coxeter graphs of type B and D. This generalizes Kauffman's diagram calculus for the Temperley–Lieb algebra.

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21

Okniński, Jan, and Magdalena Wiertel. "Combinatorics and structure of Hecke–Kiselman algebras." Communications in Contemporary Mathematics 22, no. 07 (June 15, 2020): 2050022. http://dx.doi.org/10.1142/s0219199720500224.

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Hecke–Kiselman monoids [Formula: see text] and their algebras [Formula: see text], over a field [Formula: see text], associated to finite oriented graphs [Formula: see text] are studied. In the case [Formula: see text] is a cycle of length [Formula: see text], a hierarchy of certain unexpected structures of matrix type is discovered within the monoid [Formula: see text] and this hierarchy is used to describe the structure and the properties of the algebra [Formula: see text]. In particular, it is shown that [Formula: see text] is a right and left Noetherian algebra, while it has been known that it is a PI-algebra of Gelfand–Kirillov dimension one. This is used to characterize all Noetherian algebras [Formula: see text] in terms of the graphs [Formula: see text]. The strategy of our approach is based on the crucial role played by submonoids of the form [Formula: see text] in combinatorics and structure of arbitrary Hecke–Kiselman monoids [Formula: see text].

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22

JUYUMAYA, J., and S. LAMBROPOULOU. "AN INVARIANT FOR SINGULAR KNOTS." Journal of Knot Theory and Its Ramifications 18, no. 06 (June 2009): 825–40. http://dx.doi.org/10.1142/s0218216509007324.

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In this paper we introduce a Jones-type invariant for singular knots, using a Markov trace on the Yokonuma–Hecke algebras Y d,n(u) and the theory of singular braids. The Yokonuma–Hecke algebras have a natural topological interpretation in the context of framed knots. Yet, we show that there is a homomorphism of the singular braid monoid SBn into the algebra Y d,n(u). Surprisingly, the trace does not normalize directly to yield a singular link invariant, so a condition must be imposed on the trace variables. Assuming this condition, the invariant satisfies a skein relation involving singular crossings, which arises from a quadratic relation in the algebra Y d,n(u).

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23

Pallikaros, C. A. "A note on the representation theory of the Hecke algebra of type F4." Glasgow Mathematical Journal 39, no. 1 (January 1997): 43–50. http://dx.doi.org/10.1017/s001708950003189x.

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In [4] Dipper and James investigated the representation theory of Hecke algebras of type Bn, H(Bn). Using the results in [4] and exploiting the fact that the Hecke algebra of type F4, denoted by H(W), contains two copies of H(B3) certain right ideals of H(W) will be constructed in this paper. These right ideals will be proved to be irreducible on the assumption that H(W) is semisimple.

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24

Rostam, Salim. "Cyclotomic quiver Hecke algebras and Hecke algebra of $G(r,p,n)$." Transactions of the American Mathematical Society 371, no. 6 (November 16, 2018): 3877–916. http://dx.doi.org/10.1090/tran/7485.

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25

Halverson, Tom, and Arun Ram. "Murnaghan-Nakayama Rules for Characters of Iwahori-Hecke Algebras of the Complex Reflection Groups G(r, p, n)." Canadian Journal of Mathematics 50, no. 1 (February 1, 1998): 167–92. http://dx.doi.org/10.4153/cjm-1998-009-x.

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AbstractIwahori-Hecke algebras for the infinite series of complex reflection groups G(r, p, n) were constructed recently in the work of Ariki and Koike [AK], Broué andMalle [BM], and Ariki [Ari]. In this paper we give Murnaghan-Nakayama type formulas for computing the irreducible characters of these algebras. Our method is a generalization of that in our earlier paper [HR] in whichwe derivedMurnaghan-Nakayama rules for the characters of the Iwahori-Hecke algebras of the classical Weyl groups. In both papers we have been motivated by C. Greene [Gre], who gave a new derivation of the Murnaghan-Nakayama formula for irreducible symmetric group characters by summing diagonal matrix entries in Young's seminormal representations. We use the analogous representations of the Iwahori-Hecke algebra of G(r, p, n) given by Ariki and Koike [AK] and Ariki [Ari].

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26

Fratila, Dragos. "Cusp eigenforms and the hall algebra of an elliptic curve." Compositio Mathematica 149, no. 6 (March 4, 2013): 914–58. http://dx.doi.org/10.1112/s0010437x12000784.

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AbstractWe give an explicit construction of the cusp eigenforms on an elliptic curve defined over a finite field, using the theory of Hall algebras and the Langlands correspondence for function fields and ${\mathrm{GL} }_{n} $. As a consequence we obtain a description of the Hall algebra of an elliptic curve as an infinite tensor product of simpler algebras. We prove that all these algebras are specializations of a universal spherical Hall algebra (as defined and studied by Burban and Schiffmann [On the Hall algebra of an elliptic curve I, Preprint (2005), arXiv:math/0505148 [math.AG]] and Schiffmann and Vasserot [The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials, Compositio Math. 147 (2011), 188–234]).

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27

UENO, Kimio, and Youichi SHIBUKAWA. "CHARACTER TABLE OF HECKE ALGEBRA OF TYPE AN-1 AND REPRESENTATIONS OF THE QUANTUM GROUP Uq(gln+1)." International Journal of Modern Physics A 07, supp01b (April 1992): 977–84. http://dx.doi.org/10.1142/s0217751x92004130.

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A q-analogue of the Frobenius formula is proved by means of the quantum groups Uq(gln+1), Aq(GLn+1) and Iwahori's Hecke algebra of type AN-1, and then, the character table of this Hecke algebra is investigated.

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28

Rosso, Daniele. "The mirabolic Hecke algebra." Journal of Algebra 405 (March 2014): 179–212. http://dx.doi.org/10.1016/j.jalgebra.2014.02.008.

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29

Doty, Stephen, Karin Erdmann, and Anne Henke. "A generic algebra associated to certain Hecke algebras." Journal of Algebra 278, no. 2 (August 2004): 502–31. http://dx.doi.org/10.1016/j.jalgebra.2004.04.007.

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30

Chlouveraki, Maria, and Vincent Sécherre. "The affine Yokonuma–Hecke algebra and the pro-$p$-Iwahori–Hecke algebra." Mathematical Research Letters 23, no. 3 (2016): 707–18. http://dx.doi.org/10.4310/mrl.2016.v23.n3.a7.

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31

Dunkl, Charles F. "The Classification of All Singular Nonsymmetric Macdonald Polynomials." Axioms 11, no. 5 (April 29, 2022): 208. http://dx.doi.org/10.3390/axioms11050208.

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The affine Hecke algebra of type A has two parameters q,t and acts on polynomials in N variables. There are two important pairwise commuting sets of elements in the algebra: the Cherednik operators and the Jucys–Murphy elements whose simultaneous eigenfunctions are the nonsymmetric Macdonald polynomials, and basis vectors of irreducible modules of the Hecke algebra, respectively. For certain parameter values, it is possible for special polynomials to be simultaneous eigenfunctions with equal corresponding eigenvalues of both sets of operators. These are called singular polynomials. The possible parameter values are of the form qm=t−n with 2≤n≤N. For a fixed parameter, the singular polynomials span an irreducible module of the Hecke algebra. Colmenarejo and the author (SIGMA 16 (2020), 010) showed that there exist singular polynomials for each of these parameter values, they coincide with specializations of nonsymmetric Macdonald polynomials, and the isotype (a partition of N) of the Hecke algebra module is dn−1,n−1,…,n−1,r for some d≥1. In the present paper, it is shown that there are no other singular polynomials.

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32

Hyodo, Fumitake. "A formal power series of a Hecke ring associated with the Heisenberg lie algebra over ℤp." International Journal of Number Theory 11, no. 08 (November 5, 2015): 2305–23. http://dx.doi.org/10.1142/s1793042115501055.

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This paper studies a formal power series with coefficients in a Hecke ring associated with the Heisenberg Lie algebra. We relate the series to the classical Hecke series defined by Hecke, and prove that the series has a property similar to the rationality theorem of the classical Hecke series. And then, our results recover the rationality theorem of the classical Hecke series.

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33

Moy, Allen. "Distribution Algebras on p-adic Groups and Lie Algebras." Canadian Journal of Mathematics 63, no. 5 (October 18, 2011): 1137–60. http://dx.doi.org/10.4153/cjm-2011-025-3.

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Abstract When F is a p-adic field, and is the group of F-rational points of a connected algebraic F-group, the complex vector space of compactly supported locally constant distributions on G has a natural convolution product that makes it into a ℂ-algebra (without an identity) called the Hecke algebra. The Hecke algebra is a partial analogue for p-adic groups of the enveloping algebra of a Lie group. However, has drawbacks such as the lack of an identity element, and the process is not a functor. Bernstein introduced an enlargement . The algebra consists of the distributions that are left essentially compact. We show that the process is a functor. If is a morphism of p-adic groups, let be the morphism of ℂ-algebras. We identify the kernel of in terms of Ker. In the setting of p-adic Lie algebras, with g a reductive Lie algebra, m a Levi, and the natural projection, we show that maps G-invariant distributions on to NG(m)-invariant distributions on m. Finally, we exhibit a natural family of G-invariant essentially compact distributions on g associated with a G-invariant non-degenerate symmetric bilinear form on g and in the case of SL(2) show how certain members of the family can be moved to the group.

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34

Elrifai, Elsayed. "Basis of Hecke algebras - associated to Coxeter groups - via matrices of inversion for permutations." JOURNAL OF ADVANCES IN MATHEMATICS 12, no. 4 (May 30, 2016): 6127–32. http://dx.doi.org/10.24297/jam.v12i4.346.

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Applying the matrices of inversion for permutations, we show that every element of S_{n} associates a unique canonical word in the Hecke algebra H_{n-1}(z). That provides an effective and simple algorithm for counting a linear basis of Hecke algebra H_{n}, as binary matrices.

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35

Kaliszewski, S., Magnus B. Landstad, and John Quigg. "Hecke C*-Algebras, Schlichting Completions and Morita Equivalence." Proceedings of the Edinburgh Mathematical Society 51, no. 3 (October 2008): 657–95. http://dx.doi.org/10.1017/s0013091506001419.

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AbstractThe Hecke algebra of a Hecke pair (G, H) is studied using the Schlichting completion (Ḡ, ), which is a Hecke pair whose Hecke algebra is isomorphic to and which is topologized so that is a compact open subgroup of Ḡ. In particular, the representation theory and C*-completions of are addressed in terms of the projection using both Fell's and Rieffel's imprimitivity theorems and the identity . An extended analysis of the case where H is contained in a normal subgroup of G (and in particular the case where G is a semi-direct product) is carried out, and several specific examples are analysed using this approach.

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36

Adin, Ron M., Alexander Postnikov, and Yuval Roichman. "Hecke Algebra Actions on the Coinvariant Algebra." Journal of Algebra 233, no. 2 (November 2000): 594–613. http://dx.doi.org/10.1006/jabr.2000.8441.

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37

Hardt, Andrew, Jared Marx-Kuo, Vaughan McDonald, John M. O’Brien, and Alexander Vetter. "Characters of Renner monoids and their Hecke algebras." International Journal of Algebra and Computation 30, no. 07 (August 28, 2020): 1505–35. http://dx.doi.org/10.1142/s0218196720500514.

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This paper gives a general algorithm for computing the character table of any Renner monoid Hecke algebra, by adapting and generalizing techniques of Solomon used to study the rook monoid. The character table of the Hecke algebra of the rook monoid (i.e. the Cartan type [Formula: see text] Renner monoid) was computed earlier by Dieng et al. [2], using different methods. Our approach uses analogues of so-called A- and B-matrices of Solomon. In addition to the algorithm, we give explicit combinatorial formulas for the A- and B-matrices in Cartan type [Formula: see text] and use them to obtain an explicit description of the character table for the type [Formula: see text] Renner monoid Hecke algebra.

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38

Francis, Andrew, and Lenny Jones. "On the square root of the centre of the Hecke algebra of type A." Journal of the Australian Mathematical Society 82, no. 2 (April 2007): 209–20. http://dx.doi.org/10.1017/s1446788700016037.

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AbstractIn this paper we investigate non-central elements of the Iwahori-Hecke algebra of the symmetric group whose squares are central. In particular, we describe a commutative subalgebra generated by certain non-central square roots of central elements, and the generic existence of a rank-three submodule of the Hecke algebra contained in the square root of the centre, but not in the centre. The generators for this commutative subalgebra include the longest word and elements related to trivial and sign characters of the Hecke algebra. We find elegant expressions for the squares of such generators in terms of both the minimal basis of the centre and the elementary symmetric functions of Murphy elements.

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39

DU, JIE, and JINKUI WAN. "THE QUEER -SCHUR SUPERALGEBRA." Journal of the Australian Mathematical Society 105, no. 3 (February 2, 2018): 316–46. http://dx.doi.org/10.1017/s1446788717000337.

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As a natural generalisation of $q$-Schur algebras associated with the Hecke algebra ${\mathcal{H}}_{r,R}$ (of the symmetric group), we introduce the queer $q$-Schur superalgebra associated with the Hecke–Clifford superalgebra ${\mathcal{H}}_{r,R}^{\mathsf{c}}$, which, by definition, is the endomorphism algebra of the induced ${\mathcal{H}}_{r,R}^{\mathsf{c}}$-module from certain $q$-permutation modules over ${\mathcal{H}}_{r,R}$. We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base-change property for them. We will also identify the queer $q$-Schur superalgebras with the quantum queer Schur superalgebras investigated in the context of quantum queer supergroups and provide a constructible classification of their simple polynomial representations over a certain extension of the field $\mathbb{C}(\mathbf{v})$ of complex rational functions.

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40

Hida, Haruzo. "Transcendence of Hecke operators in the big Hecke algebra." Duke Mathematical Journal 163, no. 9 (June 2014): 1655–81. http://dx.doi.org/10.1215/00127094-2690478.

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41

Rouquier, Raphaël. "Quiver Hecke Algebras and 2-Lie Algebras." Algebra Colloquium 19, no. 02 (May 3, 2012): 359–410. http://dx.doi.org/10.1142/s1005386712000247.

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We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting monoidal categories, which have a geometric description via quiver varieties, in certain cases. We present basic properties of 2-representations and describe simple 2-representations, via cyclotomic quiver Hecke algebras, and through microlocalized quiver varieties.

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Kang, Seok-Jin, Masaki Kashiwara, Myungho Kim, and Se-jin Oh. "Simplicity of heads and socles of tensor products." Compositio Mathematica 151, no. 2 (November 26, 2014): 377–96. http://dx.doi.org/10.1112/s0010437x14007799.

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AbstractWe prove that, for simple modules $M$ and $N$ over a quantum affine algebra, their tensor product $M\otimes N$ has a simple head and a simple socle if $M\otimes M$ is simple. A similar result is proved for the convolution product of simple modules over quiver Hecke algebras.

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43

Fauser, Bertfried. "Hecke algebra representations within Clifford geometric algebras of multivectors." Journal of Physics A: Mathematical and General 32, no. 10 (January 1, 1999): 1919–36. http://dx.doi.org/10.1088/0305-4470/32/10/010.

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44

Geck, Meinolf. "Leading coefficients and cellular bases of Hecke algebras." Proceedings of the Edinburgh Mathematical Society 52, no. 3 (September 23, 2009): 653–77. http://dx.doi.org/10.1017/s0013091508000394.

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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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GECK, MEINOLF. "Some applications of CHEVIE to the theory of algebraic groups." Carpathian Journal of Mathematics 27, no. 1 (2011): 64–94. http://dx.doi.org/10.37193/cjm.2011.01.07.

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The computer algebra system CHEVIE is designed to facilitate computations with various combinatorial structures arising in Lie theory, like finite Coxeter groups and Hecke algebras. We discuss some recent examples where CHEVIE has been helpful in the theory of algebraic groups, in questions related to unipotent classes, the Springer correspondence and Lusztig families.

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46

Gouveˆa, Fernando Q. "On the ordinary Hecke algebra." Journal of Number Theory 41, no. 2 (June 1992): 178–98. http://dx.doi.org/10.1016/0022-314x(92)90119-a.

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47

Schiffmann, O., and E. Vasserot. "The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials." Compositio Mathematica 147, no. 1 (July 7, 2010): 188–234. http://dx.doi.org/10.1112/s0010437x10004872.

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AbstractWe exhibit a strong link between the Hall algebra HX of an elliptic curve X defined over a finite field 𝔽l (or, more precisely, its spherical subalgebra U+X) and Cherednik’s double affine Hecke algebras $\ddot {\mathbf {H}}_n$ of type GLn, for all n. This allows us to obtain a geometric construction of the Macdonald polynomials Pλ(q,t−1) in terms of certain functions (Eisenstein series) on the moduli space of semistable vector bundles on the elliptic curve X.

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DUCHAMP, GÉRARD, FLORENT HIVERT, and JEAN-YVES THIBON. "NONCOMMUTATIVE SYMMETRIC FUNCTIONS VI: FREE QUASI-SYMMETRIC FUNCTIONS AND RELATED ALGEBRAS." International Journal of Algebra and Computation 12, no. 05 (October 2002): 671–717. http://dx.doi.org/10.1142/s0218196702001139.

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This article is devoted to the study of several algebras related to symmetric functions, which admit linear bases labelled by various combinatorial objects: permutations (free quasi-symmetric functions), standard Young tableaux (free symmetric functions) and packed integer matrices (matrix quasi-symmetric functions). Free quasi-symmetric functions provide a kind of noncommutative Frobenius characteristic for a certain category of modules over the 0-Hecke algebras. New examples of indecomposable Hn(0)-modules are discussed, and the homological properties of Hn(0) are computed for small n. Finally, the algebra of matrix quasi-symmetric functions is interpreted as a convolution algebra.

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COX, ANTON, and ALISON PARKER. "HOMOMORPHISMS AND HIGHER EXTENSIONS FOR SCHUR ALGEBRAS AND SYMMETRIC GROUPS." Journal of Algebra and Its Applications 04, no. 06 (December 2005): 645–70. http://dx.doi.org/10.1142/s0219498805001460.

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This paper surveys, and in some cases generalizes, many of the recent results on homomorphisms and the higher Ext groups for q-Schur algebras and for the Hecke algebra of type A. We review various results giving isomorphisms between Ext groups in the two categories, and discuss those cases where explicit results have been determined.

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50

Kohlhaase, Jan. "On the Iwasawa theory of the Lubin–Tate moduli space." Compositio Mathematica 149, no. 5 (February 26, 2013): 793–839. http://dx.doi.org/10.1112/s0010437x12000723.

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AbstractWe study the affine formal algebra$R$of the Lubin–Tate deformation space as a module over two different rings. One is the completed group ring of the automorphism group$\Gamma $of the formal module of the deformation problem, the other one is the spherical Hecke algebra of a general linear group. In the most basic case of height two and ground field$\mathbb {Q}_p$, our structure results include a flatness assertion for$R$over the spherical Hecke algebra and allow us to compute the continuous (co)homology of$\Gamma $with coefficients in $R$.

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