#Algebras appearing in representation theory and combinatorics
#The Hecke algebra
Let \((W,S)\) be a Coxeter system, with length function \(\ell\) and identity element \(e.\) The Iwahori–Hecke algebra \(\mathcal{H}_W\) is a deformation of the group algebra \(\setZ[W].\) One common normalization defines \(\mathcal{H}_W\) over \(\setZ[v,v^{-1}]\) with generators \(H_s,\) \(s\in S,\) satisfying the braid relations of \(W\) and the quadratic relation \[(H_s-v)(H_s+v^{-1})=0.\] For each \(w\in W,\) the products along reduced words define a standard basis \(\{H_w : w\in W\}.\) At \(v=1,\) this specializes back to the group algebra.
J. Gruber introduces pseudo-centralizer subalgebras of affine Hecke algebras [Gru26]. The construction starts inside the double affine Hecke algebra and gives a subalgebra \(\mathcal{B}\) of the affine Hecke algebra which can be viewed as a \(v\)-deformation of Lam’s affine Fomin–Stanley subalgebra, a combinatorial model for the affine Grassmannian Schur-basis story. In type \(A\) and in types \(B_2\) and \(G_2,\) Gruber constructs a standard basis indexed by minimal finite-Weyl-group coset representatives in the affine Weyl group, and then a canonical basis defined using the Kazhdan–Lusztig basis. The paper also formulates positivity conjectures for this canonical basis and relates the construction to the center of the affine Hecke algebra. H. Murata proves a quantum imaginary Schur–Weyl duality for quiver Hecke algebras of untwisted affine type \(A\) [Mur26]. For arbitrary quiver-Hecke parameters, the resulting parameter \(t\) determines an Iwahori–Hecke algebra of the symmetric group, and the endomorphism algebras in the imaginary strata are identified with these Hecke algebras. The same framework computes characters of simple modules using dual canonical bases and Kazhdan–Lusztig polynomials, and identifies standard-module characters with PBW vectors under semisimplicity hypotheses.
#Kazhdan–Lusztig polynomials
D. Kazhdan and G. Lusztig introduced Kazhdan–Lusztig polynomials in their study of Coxeter groups and Hecke algebras [KL79]. They are polynomials \(P_{u,w}(q)\) indexed by pairs \(u,w\in W,\) with \(P_{u,w}(q)=0\) unless \(u\leq w\) in Bruhat order and \(P_{w,w}(q)=1.\)
The polynomials arise as transition coefficients between the standard basis of \(\mathcal{H}_W\) and the self-dual Kazhdan–Lusztig basis. They encode deep geometric and representation-theoretic information; for Weyl groups they are closely related to intersection cohomology of Schubert varieties and to characters in category \(\mathcal{O}.\)
Several variants occur elsewhere on this site, including parabolic Kazhdan–Lusztig polynomials in the Schur expansion of LLT polynomials, and matroid Kazhdan–Lusztig polynomials in matroid theory.
#The \(q\)-Klyachko algebra
The \(q\)-Klyachko algebra was introduced by P. Nadeau and V. Tewari [NT25]. It is a \(q\)-deformation of the \(\symS_n\)-invariant part of the rational cohomology ring of the permutahedral variety, designed to interpolate between Klyachko’s formula and I. Macdonald’s reduced-word formula.
It is given by \[\setQ[\dotsc,u_{-1},u_0,1,u_2,\dotsc] / \langle (q+1)u_i^2 = q u_i u_{i-1} + u_i u_{i+1} \rangle.\]
#Cluster algebras
Cluster algebras are commutative algebras generated from overlapping clusters by mutation, introduced by S. Fomin and A. Zelevinsky [FZ02].
#Hopf algebras
Hopf algebras are algebras equipped with compatible coproduct, counit, and antipode maps; for a combinatorial survey, see D. Grinberg and V. Reiner [GR14].
Bibliography
- [FZ02]Sergey Fomin and Andrei Zelevinsky. Cluster algebras I: foundations. Journal of the American Mathematical Society, 15(2):497–529, 2002.
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@article{FominZelevinsky2002, author = {Sergey Fomin and Andrei Zelevinsky}, title = {Cluster algebras {I}: Foundations}, journal = {Journal of the American Mathematical Society}, volume = {15}, number = {2}, pages = {497--529}, year = {2002}, doi = {10.1090/S0894-0347-01-00385-X}, eprint = {math/0104151}, archivePrefix = {arXiv} } - [GR14]Darij Grinberg and Victor Reiner. Hopf algebras in combinatorics. arXiv:1409.8356, 2014. Lecture notes
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@misc{GrinbergReiner2014Hopf, author = {Darij Grinberg and Victor Reiner}, title = {Hopf Algebras in Combinatorics}, year = {2014}, eprint = {1409.8356}, archivePrefix = {arXiv}, primaryClass = {math.CO}, note = {Lecture notes} } - [Gru26]Jonathan Gruber. Pseudo-centralizers in affine Hecke algebras. arXiv:2607.00426, 2026.
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@article{Gruber2026x, author = {Jonathan Gruber}, title = {Pseudo-centralizers in affine {H}ecke algebras}, year = {2026}, eprint = {2607.00426}, url = {https://arxiv.org/abs/2607.00426}, journal = {arXiv e-prints} } - [KL79]David Kazhdan and George Lusztig. Representations of Coxeter groups and Hecke algebras. Inventiones Mathematicae, 53(2):165–184, June 1979.
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@article{KazhdanLusztig1979, doi = {10.1007/BF01390031}, url2 = {https://doi.org/10.1007/BF01390031}, year = {1979}, month = jun, publisher = {Springer New York}, volume = {53}, number = {2}, pages = {165--184}, author = {David Kazhdan and George Lusztig}, title = {Representations of {C}oxeter groups and {H}ecke algebras}, journal = {Inventiones Mathematicae} } - [Mur26]Haruto Murata. Quantum imaginary Schur-Weyl duality. arXiv:2607.01026, 2026.
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@article{Murata2026x, author = {Haruto Murata}, title = {Quantum imaginary {S}chur-{W}eyl duality}, year = {2026}, eprint = {2607.01026}, url = {https://arxiv.org/abs/2607.01026}, journal = {arXiv e-prints} } - [NT25]Philippe Nadeau and Vasu Tewari. A $q$-deformation of an algebra of Klyachko, and Macdonald’s reduced word formula. Transactions of the American Mathematical Society, 378(12):8821–8870, 2025.
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@article{NadeauTewari2025QKlyachko, author = {Philippe Nadeau and Vasu Tewari}, title = {A $q$-deformation of an algebra of {K}lyachko, and {M}acdonald's reduced word formula}, year = {2025}, journal = {Transactions of the American Mathematical Society}, volume = {378}, number = {12}, pages = {8821--8870}, doi = {10.1090/tran/9481}, url = {https://doi.org/10.1090/tran/9481}, eprint = {2106.03828} }