Applications of Hecke Algebras: Representations, Knots and Physics

ANR project JCJC ANR-18-CE40-0001

Period

From January 2019 to December 2023

Summary

Between representation theory, knot theory and integrable systems, interactions are plentiful. This is one of the several fruitful (and two-way) bridges between mathematics and theoretical physics. If one had to identify a common ground for these interactions, for example in the sense of a common algebraic structure, quantum groups would certainly be a good answer. Hecke algebras, considered in a large sense, could be another.

Hecke algebras originally appeared in the theory of modular forms in the 30's. Since then the name "Hecke algebras" has been progressively used to refer to a wide variety of objects, appearing and extensively studied in several areas of mathematics. Classes of examples of Hecke algebras of special interest for this project are:
• Centralisers (endomorphisms algebras) of induced representations;
• Deformations of Coxeter groups and (complex) reflection groups;
• Quotients of group algebras of (generalised) braid groups;
• Centralisers of tensor representations of quantum groups.

Quite remarkably, all the classes of examples above have a lot in common and this is the main reason why Hecke algebras are so important in modern mathematics: they can be studied from many points of view and they have applications in many different fields.
The project is centered on the study of Hecke algebras and concerns their applications to/interplay between different areas of mathematics and physics. We will focus on three main areas, where Hecke algebras and related algebras play an important role:
• Representation theory of different kind of Hecke algebras (and generalisations);
• Low-dimensional topology, mainly the study of braid groups and invariants of links;
• Theoretical physics (integrable systems, statistical models, quantum field theory).

One major objective of this project is to study these three thematics simultaneously, especially focusing on interactions between them, and considering Hecke algebras as bridges between these areas.

Participants

• Maria Chlouveraki (Laboratoire de Mathématiques de Versailles)
• Nicolas Crampé (CNRS, Institut Denis Poisson de Tours)
• Azat Gainutdinov (CNRS, Institut Denis Poisson de Tours)
• Jérémie Guilhot (Institut Denis Poisson de Tours)
• Nicolas Jacon (Laboratoire de Mathématiques de Reims)
• Loïc Poulain d'Andecy (Coordinator) (Laboratoire de Mathématiques de Reims)
• Emmanuel Wagner (Institut Mathématiques de Jussieu de Paris)

Events

• February 2019: Réunion de lancement, Reims. Orateurs: A. Gainutdinov, L. Poulain d'Andecy, E. Wagner
• March 2019: Winter Braids IX , March, 4th - 7th 2019
• February 2020: Winter Braids X , March, 17th - 21st 2020
• June 2021: Deuxième réunion de l'ANR, Paris. Orateurs: N. Crampé, N. Jacon, J. Guilhot, B. Rognerud
• November 2022: Troisième réunion de l'ANR, Tours. Orateurs: P. Baseilhac, A. Gainutdinov, L. Poulain d'Andecy
• March 2023: Winter school of the ANR project, "Applications of Hecke and related algebras: Representations, Integrability and Physics", Les Houches, February 27th - 3rd March 2023
• July 2023: Final conference of the ANR project, "Hecke algebras and applications", Spetses, 16th - 21st July 2023
• December 2023: Réunion finale de l'ANR, Paris. Orateurs: M. Chlouveraki, E. Wagner

Selected publications

• A. Lacabanne, L. Poulain d'Andecy, Framization of Schur-Weyl duality and Yokonuma-Hecke type algebras, preprint arXiv:2312.14796
• A. Lafay, A. M. Gainutdinov, J. L. Jacobsen, Integrability of rank-two web models, preprint arXiv:2311.14805
• N. Crampé, J. Gaboriaud, L. Poulain d'Andecy and L. Vinet, Matrix elements of SO(3) in sl_3 representations as bispectral multivariate functions, Journal of Mathematical Physics 64 (2023), no. 11, Paper No. 111702, 16pp. arXiv:2308.12809
• J. Berger, A. M. Gainutdinov, I. Runkel, Non-semisimple link and manifold invariants for symplectic fermions, preprint arXiv:2307.06069
• N. Jacon, C.Lecouvey, Mullineux involution and crystal isomorphisms II, preprint arXiv:2307.01065
• J. Guilhot, C. Lecouvey, P. Tarrago, Homology rings of affine grassmannians and positively multiplicative graphs, preprint arXiv:2306.16105
• L. Poulain d'Andecy and M. Zaimi, Fused Hecke algebra and one-boundary algebras, to appear in Pacific Journal of Mathematics. arXiv:2306.10937
• Y. Qi, J. Sussan, E. Wagner, Symmetries of equivariant Khovanov-Rozansky homology, preprint arXiv:2306.10729
• Q. Labriet and L. Poulain d'Andecy, Identities for Rankin-Cohen brackets, Racah coefficients and associativity, to appear in Letters in Mathematical Physics. arXiv:2304.10803
• Q. Labriet and L. Poulain d'Andecy, Realisations of Racah algebras using Jacobi operators and convolution identities, Advances in Applied Mathematics 153 (2024), Paper No. 102620, 31pp. arXiv:2304.00820
• N. Crampé, L. Frappat, L. Poulain d'Andecy and E. Ragoucy, The higher-rank Askey-Wilson algebra and its braid group automorphisms, SIGMA Symmetry Integrability and Geometry. Methods and Applications 19 (2023) Paper No. 077, 36pp. arXiv:2303.17677
• P. Baseilhac, A. M. Gainutdinov, G. Lemarthe, Fused K-operators and the q-Onsager algebra, preprint arXiv:2301.00781
• P.-A. Bernard, N. Crampé, L. Poulain d'Andecy, L. Vinet and M. Zaimi, Bivariate P-polynomial association schemes, to appear in Algebraic Combinatorics. arXiv:2212.10824
• J. Guilhot, E. Little, J. Parkinson, On J-folded alcove paths and combinatorial representations of affine Hecke algebras, preprint arXiv:2212.10781.
• Y. Qi, J. Sussan, E. Wagner, Symmetries of gl(N)-foams, preprint arXiv:2212.10106.
• D. Chernyak, A. M. Gainutdinov, J. L. Jacobsen, H. Saleur, Algebraic Bethe Ansatz for the open XXZ spin chain with non-diagonal boundary terms via Uqsl2 symmetry, SIGMA Symmetry Integrability Geom. Methods Appl. 19 (2023), Paper No. 046, 47 pp. arXiv:2212.09696
• J. Guilhot, C. Lecouvey, P. Tarrago, Quantum cohomology of the Grassmannian and unitary Dyson Brownian motion, preprint arXiv:2211.12836.
• A. Beliakova, K. Putyra, E. Wagner, A proof of Dunfield-Gukov-Rasmussen conjecture, preprint arXiv:2210.00878.
• D. Chernyak, A. M. Gainutdinov, H. Saleur, Uqsl2-invariant non-compact boundary conditions for the XXZ spin chain, J. High Energy Phys.(2022), no. 11, Paper No. 16, 64 pp. arXiv:2207.12772
• N. Crampé, L. Poulain d'Andecy, L. Vinet and M. Zaimi, Askey-Wilson braid algebra and centralizer of U_q(sl_2), Annales de l'Institut Henri Poincaré 24 (2023), no. 6, 1897-1922. arXiv:2206.11150
• N. Crampé, L. Frappat, E. Ragoucy, Representations of the rank two Racah algebra and orthogonal multivariate polynomials, Linear Algebra and its Applications 664 (2023) 165-215. arXiv:2206.01031
• P.-A. Bernard, N. Crampé, R. Nepomechie, G. Parez, L. Poulain d'Andecy and L. Vinet, Entanglement of inhomogeneous free fermions on hyperplane lattices, Nuclear Physics B 984 (2022) Paper No.115975, 28pp. arXiv:2206.06509
• J. Guilhot, C. Lecouvey, P. Tarrago, Basics on positively multiplicative graphs and algebras, preprint arXiv:2205.08889.
• L. Poulain d'Andecy, Fusion for the Yang-Baxter equation and the braid group, Winter Braids Lecture Notes Vol. 7 (2020), Course III, p. 1-49. arXiv:2204.03483
• N. Jacon, A.Lacabanne, On Calogero-Moser cellular characters for imprimitive complex reflection groups, to appear in Tunisian Journal of Mathematics. arXiv:2204.01014
• Y. Qi, J. Sussan, E. Wagner, A categorification of the colored Jones polynomial at a root of unity, preprint arXiv:2111.13195.
• T. Gerber, J. Guilhot, C. Lecouvey, Generalised Howe duality and injectivity of induction: the symplectic case, Comb. Theory, 2(2):42 (2022). arXiv:2110.04029
• N. Crampé, L. Poulain d'Andecy and L. Vinet, The missing label of su_3 and its symmetry, Communications in Mathematical Physics 400 (2023), no. 1, 179-213. arXiv:2110.03521
• A. Lafay, A. M. Gainutdinov, J. L. Jacobsen, Uq(sl3) web models: Locality, phase diagram and geometrical defects, Nuclear Phys. B 979 (2022), Paper No. 115789, 59 pp. arXiv:2107.10106
• N. Jacon, Mullineux involution and crystal isomorphisms I, Combinatorial Theory, 3(2), 2023. arXiv:2107.02641
• M. Chlouveraki, N. Jacon, Defect in cyclotomic Hecke algebras, Math. Z. 305, 57 (2023). arXiv:2105.08580.
• N. Crampé, J. Gaboriaud, L. Poulain d'Andecy and L. Vinet, Racah algebras, the centralizer Z_n(sl_2) and its Hilbert-Poincaré series, Annales de l'Institut Henri Poincaré 23 (2022), no. 7, 2657-2682. arXiv:2105.01086
• N. Jacon, Two maps on affine type A crystals and Hecke algebras, Electronic Journal of Combinatorics, Volume 28, Issue 2 (2021), 2-21. arXiv:2102.05335
• A. Lafay, A. M. Gainutdinov, J. L. Jacobsen, Uq(sln) web models and Zn spin interfaces, J. Stat. Mech. Theory Exp.(2021), no. 5, Paper No. 053104, 33 pp. arXiv:2101.00282
• O. Brunat, J.-B. Gramain, N. Jacon, On unitriangular basic sets for symmetric and alternating groups, Journal of Algebra, 620 (2023), 257-292. arXiv:2011.00815
• M. De Renzi, A. M. Gainutdinov, N. Geer, B. Patureau-Mirand, I. Runkel, Mapping Class Group Representations From Non-Semisimple TQFTs, Commun. Contemp. Math. 25 (2023), no. 1, Paper No. 2150091, 52 pp. arXiv:2010.14852
• N. Crampé, L. Frappat, J. Gaboriaud, L. Poulain d'Andecy, E. Ragoucy and L. Vinet, The Askey-Wilson algebra and its avatars, Journal of Physics A: Mathematical and Theoretical 54 (2021) no. 6, 063001 (32pp). arXiv:2009.14815
• N. Crampé, L. Vinet, M. Zaimi, Temperley-Lieb, Birman-Murakami-Wenzl and Askey-Wilson algebras and other centralizers of Uq(sl2), Annales Henri Poincaré 22 (2021) 3499. arXiv:2008.04905
• E. Chavli, M. Chlouveraki, The freeness and trace conjectures for parabolic Hecke subalgebras, Journal of Pure and Applied Algebra 226 (2022), 107050. arXiv:2007.11535.
• N. Crampé, L. Poulain d'Andecy and L. Vinet, A Calabi-Yau algebra with E6 symmetry and the Clebsch-Gordan series of sl(3), Journal of Lie Theory 31 (2021), no. 4, 1085-1112. arXiv:2005.13444
• N. Crampé and L. Poulain d'Andecy, Baxterisation of the fused Hecke algebra and R-matrices with gl(N)-symmetry, Letters in Mathematical Physics 111 (2021), no. 4, Paper No. 92, 21 pp. arXiv:2004.05035
• N. Crampé and L. Poulain d'Andecy, Fused braids and centralisers of tensor representations of Uq(glN), Algebras and Representation Theory 26 (2023), no. 3, 901-955. arXiv:2001.11372
• N. Jacon, C. Lecouvey, Cores of Ariki-Koike algebras, Documenta Mathematica, 26 (2021). arXiv:1912.08461
• M. De Renzi, A. M. Gainutdinov, N. Geer, B. Patureau-Mirand, I. Runkel, 3-dimensional TQFTs from non-semisimple modular categories, Selecta Math. (N.S.) 28 (2022), no. 2, Paper No. 42, 60 pp. arXiv:1912.02063.
• N. Crampé, W. van de Vijver, L. Vinet, Racah problems for the oscillator algebra, the Lie algebra sln, and multivariate Krawtchouk polynomials, Annales Henri Poincaré 21 (2020) 3939. arXiv:1909.12643
• N. Jacon, C. Lecouvey, Keys and Demazure crystals for Kac-Moody algebras, Journal of Combinatorial Algebra 4 (2020) 325-358. arXiv:1909.09520
• L.-H. Robert, E. Wagner, State sums for some super quantum link invariants, Topology and Geometry: A Collection of Papers Dedicated to Vladimir G. Turaev, ed. A. Papadopoulos, 2021. arXiv:1909.02305
• N. Crampé, R.I. Nepomechie, L. Vinet, Free-Fermion entanglement and orthogonal polynomials, J. Stat. Mech. (2019) 093101. arXiv:1907.00044
• N. Crampé, L. Frappat, L. Vinet, Centralizers of the superalgebra osp(1|2): the Brauer algebra as a quotient of the Bannai-Ito algebra, J. Phys. A 52 (2019), no. 42, 424001, 11 pp. arXiv:1906.03936.
• N. Crampé, L. Poulain d'Andecy and L. Vinet, Temperley-Lieb, Brauer and Racah algebras and other centralizers of su(2), Transactions of the American Mathematical Society 373 (2020), no. 7, 4907-4932. arXiv:1905.06346
• M. Chlouveraki, D. Goundaroulis, A. Kontogeorgis, S. Lambropoulou, A generalized skein relation for Khovanov homology and a categorification of the Θ-invariant, Proceedings of the Royal Society of Edinburgh Section A: Mathematics, Vol. 151 , No. 6 (2021), 173-1757. arXiv:1904.07794
• L. Poulain d'Andecy and S. Rostam, Morita equivalences for cyclotomic Hecke algebras of type B and D, Bulletin de la Société Mathématique de France 149 (2021), no. 1, 179-233. arXiv:1903.01580
• N.Jacon, Uglov bipartitions and extended Young diagrams, Electronic Journal of Combinatorics, Vol. 26, Issue 3, (23 pages) 2019. arXiv:1903.01109
• L.-H. Robert, E. Wagner, A quantum categorification of the Alexander polynomial, Geometry and Topology 26 (2022), no. 5, 1985-2064. arXiv:1902.05648
• L. Poulain d'Andecy and R. Walker, Affine Hecke algebras of type D and generalisations of quiver Hecke algebras, Journal of Algebra 552 (2020), 1-37. arXiv:1901.09978

Contact

Loïc Poulain d'Andecy

   loic.poulain-dandecy(at)univ-reims.fr

Laboratoire de Mathématiques de Reims (LMR) - UMR 9008
       U.F.R. Sciences Exactes et Naturelles
       Moulin de la Housse - BP 1039
       51687 REIMS cedex 2 FRANCE