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#Algebras appearing in representation theory and combinatorics

#The coinvariant algebra

Let \(k\) be a field of characteristic zero. The symmetric group \(\symS_n\) acts on \(k[x_1,\dotsc,x_n]\) by permuting variables. Its coinvariant algebra is the graded quotient \[R_n\coloneqq k[x_1,\dotsc,x_n]/\langle e_1,\dotsc,e_n\rangle,\] where \(e_i=e_i(x_1,\dotsc,x_n)\) is the \(i\)th elementary symmetric polynomial. Equivalently, we quotient by the ideal generated by all symmetric polynomials with zero constant term. Giving each \(x_i\) degree one, we have \[\operatorname{Hilb}(R_n;t) =\sum_{d\geq0}\dim_k(R_n)_d\,t^d =\prod_{i=1}^n(1+t+\dotsb+t^{i-1}).\] In particular, \(\dim_k R_n=n!,\) and, after forgetting the grading, \(R_n\) is the regular representation of \(\symS_n\); see [HRS18, Sec. 1]. The classes of Schubert polynomials indexed by \(\symS_n\) form a basis. Over \(\setC,\) this is the cohomology ring of the complete flag variety, with polynomial degree \(d\) corresponding to cohomological degree \(2d\); see [Ful97]. The diagonal coinvariant ring uses two sets of variables and the diagonal symmetric-group action instead.

#NilCoxeter and nilHecke algebras

For a Coxeter system \((W,S),\) the nilCoxeter algebra over \(k\) has generators \(u_s,\) \(s\in S,\) satisfying the braid relations and \(u_s^2=0.\) It has a basis \(\{u_w:w\in W\},\) defined by products along reduced words, with multiplication \[u_wu_v= \begin{cases} u_{wv},&\ell(wv)=\ell(w)+\ell(v),\\ 0,&\text{otherwise}, \end{cases}\] where \(\ell\) is Coxeter length. For \(W=\symS_n,\) its generators \(u_1,\dotsc,u_{n-1}\) satisfy \[u_i u_{i+1}u_i=u_{i+1}u_i u_{i+1},\qquad u_i u_j=u_j u_i\quad(|i-j|\gt{}1).\] The affine version and its commuting Pieri operators appear in the affine Schur-function construction.

The type \(A\) nilHecke algebra is the subalgebra of \(\operatorname{End}_k(k[x_1,\dotsc,x_n])\) generated by multiplication by the \(x_j\) and the divided difference operators \[\partial_i f=\frac{f-s_i f}{x_i-x_{i+1}}, \qquad 1\leq i\lt{}n,\] where \(s_i\) exchanges \(x_i\) and \(x_{i+1}.\) The \(\partial_i\) generate a copy of the nilCoxeter algebra. The mixed relations, with \(x_j\) now denoting multiplication operators, include \[\partial_i x_i-x_{i+1}\partial_i=1, \qquad x_i\partial_i-\partial_i x_{i+1}=1,\] and \(\partial_i x_j=x_j\partial_i\) for \(j\notin\{i,i+1\}.\) These operators also act on the coinvariant algebra and provide the operator description of Schubert polynomials; see [KK86]. Unlike the 0-Hecke algebra in its idempotent normalization, the nilCoxeter generators square to zero.

#The plactic algebra

Let \(A\) be a totally ordered alphabet and \(k\) a field. The plactic algebra is the monoid algebra \(k[\operatorname{Pl}(A)]\) of the plactic monoid. Equivalently, it is the free associative algebra on the letters of \(A,\) modulo the Knuth relations. It has a basis indexed by semistandard Young tableaux with entries in \(A\): two words give the same basis element exactly when they have the same RSK insertion tableau. Multiplication concatenates representatives and takes the insertion tableau of the resulting word; see [Ful97, Ch. 2].

#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.

N. Jing and N. Liu establish quantum super Schur–Weyl duality for mirabolic Hecke algebras with \(q\) indeterminate [JL26]. They derive a super Frobenius formula in hook Schur functions, a Murnaghan–Nakayama rule, and a Roichman character formula using super mirabolic RSK.

#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 Temperley–Lieb algebra

Let \(R\) be a commutative ring, \(\delta\in R,\) and \(n\geq1.\) The Temperley–Lieb algebra \(\operatorname{TL}_n(\delta)\) is the unital associative \(R\)-algebra generated by \(U_1,\dotsc,U_{n-1}\) with relations \[\begin{aligned} U_i^2&=\delta U_i,\\ U_iU_jU_i&=U_i &&\text{if }|i-j|=1,\\ U_iU_j&=U_jU_i &&\text{if }|i-j|\gt{}1. \end{aligned}\] Its diagram basis consists of noncrossing pairings of \(n\) points on the top and \(n\) points on the bottom of a rectangle. We multiply diagrams by stacking them and replacing each closed loop by a factor of \(\delta.\) Thus the algebra is free of rank \(C_n=\frac{1}{n+1}\binom{2n}{n}\) over \(R.\) For an introduction to both presentations, see S. Abramsky’s exposition, Section 2.

With \(\delta=v+v^{-1},\) the assignment \(H_{s_i}\mapsto v-U_i\) gives a surjection from the type \(A\) Hecke algebra onto \(\operatorname{TL}_n(\delta),\) in the normalization above. On this site, related combinatorics appears in Temperley–Lieb crystals.

#Clifford algebras

Let \(V\) be a finite-dimensional vector space over a field \(k\) of characteristic different from \(2,\) and let \(Q:V\to k\) be a quadratic form. The Clifford algebra is \[\operatorname{Cl}(V,Q)\coloneqq T(V)/\langle u\otimes u-Q(u)1:u\in V\rangle,\] where \(T(V)=\bigoplus_{r\geq0}V^{\otimes r}\) is the tensor algebra. Writing \(B(u,w)=\frac12(Q(u+w)-Q(u)-Q(w)),\) the relations become \(uw+wu=2B(u,w)1.\) We use the convention \(u^2=Q(u)1\); some authors use its negative. When \(Q=0,\) this is the exterior algebra. In all cases, \(\dim_k\operatorname{Cl}(V,Q)=2^{\dim_k V}.\)

For example, over \(\setC\) with \(Q(x)=x_1^2+\dotsb+x_n^2,\) we obtain the algebra \(\operatorname{Cl}_n\) with generators \(c_1,\dotsc,c_n\) satisfying \[c_i^2=1,\qquad c_ic_j=-c_jc_i\quad(i\ne j).\] It has basis \(c_{i_1}\dotsm c_{i_r}\) with \(i_1\lt{}\dotsb\lt{}i_r,\) including \(1.\) Declaring the generators odd makes it a \(\setZ/2\setZ\)-graded algebra. The symmetric group acts by permuting the generators; the semidirect product \(\operatorname{Cl}_n\rtimes\setC[\symS_n]\) is the Sergeev algebra, also called the Hecke–Clifford algebra. It connects spin representations of symmetric groups with Schur’s \(Q\) functions; see [WW12, Secs. 2–3].

#Double affine Hecke algebras

The double affine Hecke algebra (DAHA), introduced by I. Cherednik, extends an affine Hecke algebra by a second family of commuting invertible elements; see [Che95] and P. Etingof’s overview. In the type \(GL_n\) version, it contains the finite Hecke generators \(T_1,\dotsc,T_{n-1}\) and two Laurent polynomial subalgebras generated by \(X_1^{\pm1},\dotsc,X_n^{\pm1}\) and \(Y_1^{\pm1},\dotsc,Y_n^{\pm1}.\) The \(X_i\) commute with one another, as do the \(Y_i,\) but the two families do not commute with each other. Their cross-relations depend on parameters \(q,t.\)

In the polynomial representation on Laurent polynomials, \(X_i\) acts by multiplication by \(x_i,\) while the \(Y_i\) act by commuting \(q\)-difference-reflection operators. For generic parameters, nonsymmetric Macdonald polynomials are their joint eigenfunctions. Symmetric polynomials in the \(Y_i\) preserve symmetric Laurent polynomials and have the symmetric Macdonald polynomials as joint eigenfunctions.

#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 \[\setQ[\dotsc,u_{-1},u_0,1,u_2,\dotsc] / \langle (q+1)u_i^2 = q u_i u_{i-1} + u_i u_{i+1} \rangle.\]

#Kirchhoff algebras

Let \(G=(V,E)\) be a finite connected loopless graph and let \(k\) be a field of characteristic zero. Its Kirchhoff polynomial is \[F_G(x)\coloneqq\sum_{T\text{ spanning tree of }G}\prod_{e\in T}x_e.\] Let \(k[\partial_e:e\in E]\) act on \(k[x_e:e\in E]\) by \(\partial_e=\partial/\partial x_e.\) The Kirchhoff algebra is the apolar algebra \[\operatorname{Kir}(G)\coloneqq k[\partial_e:e\in E]/\operatorname{Ann}(F_G), \qquad \operatorname{Ann}(F_G)=\{D:D(F_G)=0\}.\] It is a finite-dimensional graded Gorenstein algebra with top degree \(|V|-1.\) More generally, replacing spanning trees by the bases of a matroid gives the Kirchhoff algebra of that matroid. See [Aka26, Sec. 1] for this terminology and its connection with Lefschetz properties.

#Postnikov–Shapiro algebras

Let \(G\) be a finite connected loopless graph with vertices \(\{0,1,\dotsc,n\},\) where \(0\) is a chosen root, and work over a field \(k\) of characteristic zero. For a nonempty \(I\subseteq[n],\) let \(d_I\) be the number of edges joining \(I\) to its complement in \(\{0,1,\dotsc,n\}.\) The tree-counting and forest-counting Postnikov–Shapiro algebras have the power-ideal presentations \[\begin{aligned} B_G&\coloneqq k[x_1,\dotsc,x_n]/ \left\langle\left(\sum_{i\in I}x_i\right)^{d_I}: \varnothing\ne I\subseteq[n]\right\rangle,\\ \widehat B_G&\coloneqq k[x_1,\dotsc,x_n]/ \left\langle\left(\sum_{i\in I}x_i\right)^{d_I+1}: \varnothing\ne I\subseteq[n]\right\rangle. \end{aligned}\] Their dimensions count, respectively, spanning trees and acyclic edge subsets of \(G\) (including the empty subset); see [PS04, Secs. 3 and 11]. For \(G=K_{n+1},\) the algebra \(B_G\) has a monomial basis indexed by parking functions, and dimension \((n+1)^{n-1}.\) These vertex-variable power-ideal algebras are different constructions from the Kirchhoff apolar algebra defined using edge-variable differential operators.

#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].

F. Aliniaeifard and S. v. Willigenburg introduce rooted Hopf algebras, graded connected Hopf algebras equipped with a group-like element in their completion, and identify their category as dual to the category of combinatorial Hopf algebras [AW26]. Roots extend Bernstein and raising operators, producing analogues of Schur, Hall–Littlewood, and Macdonald functions; the construction is developed explicitly for symmetric functions in noncommuting variables.

Bibliography

  1. [Aka26]Yohji Akama. Strong Lefschetz property of the Kirchhoff algebra of the complete tripartite graph $K_{1,1,n}$. arXiv:2610.21217v3, 2026.
    .bib
    @article{Akama2026Kirchhoff,
  author = {Yohji Akama},
  title = {Strong {L}efschetz property of the {K}irchhoff algebra of the complete tripartite graph {$K_{1,1,n}$}},
  journal = {arXiv e-prints},
  year = {2026},
  eprint = {2610.21217v3},
  url = {https://arxiv.org/abs/2610.21217v3}
}
  2. [AW26]Farid Aliniaeifard and Stephanie van Willigenburg. A universal framework for Hopf algebras in combinatorics. arXiv:2610.21289v1, 2026.
    .bib
    @article{AliniaeifardWilligenburg2026x,
  author = {Farid Aliniaeifard and {Stephanie van Willigenburg}},
  title = {A universal framework for {H}opf algebras in combinatorics},
  year = {2026},
  eprint = {2610.21289v1},
  url = {https://arxiv.org/abs/2610.21289v1},
  journal = {arXiv e-prints}
}
  3. [Che95]Ivan Cherednik. Double affine Hecke algebras and Macdonald’s conjectures. Annals of Mathematics, 141(1):191–216, 1995.
    .bib
    @article{Cherednik1995,
 ISSN = {0003486X},
 URL = {http://www.jstor.org/stable/2118632},
 author = {Ivan Cherednik},
 journal = {Annals of Mathematics},
 number = {1},
 pages = {191--216},
 publisher = {Annals of Mathematics},
 title = {Double Affine {H}ecke Algebras and {M}acdonald's Conjectures},
 volume = {141},
 year = {1995}
}
  4. [FZ02]Sergey Fomin and Andrei Zelevinsky. Cluster algebras I: foundations. Journal of the American Mathematical Society, 15(2):497–529, 2002.
    .bib
    @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}
}
  5. [Ful97]William Fulton. Young tableaux: With applications to representation theory and geometry. London mathematical society student texts (book 35). Cambridge University Press, 1997.
    .bib
    @book{Fulton1997,
  title = {Young Tableaux: With Applications to Representation Theory and Geometry},
  doi = {10.1017/cbo9780511626241},
  author = {William Fulton},
  isbn = {978-0511626241},
  series = {London Mathematical Society Student Texts (Book 35)},
  year = {1997},
  publisher = {Cambridge University Press}
}
  6. [GR14]Darij Grinberg and Victor Reiner. Hopf algebras in combinatorics. arXiv:1409.8356, 2014. Lecture notes
    .bib
    @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}
}
  7. [Gru26]Jonathan Gruber. Pseudo-centralizers in affine Hecke algebras. arXiv:2607.00426, 2026.
    .bib
    @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}
}
  8. [HRS18]James Haglund, Brendon Rhoades and Mark Shimozono. Ordered set partitions, generalized coinvariant algebras, and the Delta conjecture. Advances in Mathematics, 329:851–915, 2018.
    .bib
    @article{HaglundRhoadesShimozono2018,
  author = {Haglund, James and Rhoades, Brendon and Shimozono, Mark},
  title = {Ordered set partitions, generalized coinvariant algebras, and the
    {D}elta conjecture},
  year = {2018},
  journal = {Advances in Mathematics},
  volume = {329},
  pages = {851--915},
  publisher = {Elsevier BV},
  doi = {10.1016/j.aim.2018.01.028},
  url = {http://dx.doi.org/10.1016/j.aim.2018.01.028},
  issn = {0001-8708}
}
  9. [JL26]Naihuan Jing and Ning Liu. Quantum super Schur–Weyl duality and character formulas for mirabolic Hecke algebras. arXiv:2610.11101, 2026.
    .bib
    @article{JingLiu2026Mirabolic,
  author = {Naihuan Jing and Ning Liu},
  title = {Quantum super {S}chur--{W}eyl duality and character formulas
    for mirabolic {H}ecke algebras},
  year = {2026},
  eprint = {2610.11101},
  url = {https://arxiv.org/abs/2610.11101},
  journal = {arXiv e-prints}
}
  10. [KL79]David Kazhdan and George Lusztig. Representations of Coxeter groups and Hecke algebras. Inventiones Mathematicae, 53(2):165–184, June 1979.
    .bib
    @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}
}
  11. [KK86]Bertram Kostant and Shrawan Kumar. The nil Hecke ring and cohomology of $G/P$ for a Kac–Moody group $G$. Advances in Mathematics, 62(3):187–237, 1986.
    .bib
    @article{KostantKumar1986NilHecke,
  author = {Bertram Kostant and Shrawan Kumar},
  title = {The nil {H}ecke ring and cohomology of {$G/P$} for a {K}ac--{M}oody group {$G$}},
  journal = {Advances in Mathematics},
  volume = {62},
  number = {3},
  pages = {187--237},
  year = {1986},
  doi = {10.1016/0001-8708(86)90101-5},
  url = {https://doi.org/10.1016/0001-8708(86)90101-5}
}
  12. [Mur26]Haruto Murata. Quantum imaginary Schur-Weyl duality. arXiv:2607.01026, 2026.
    .bib
    @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}
}
  13. [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.
    .bib
    @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}
}
  14. [PS04]Alexander Postnikov and Boris Shapiro. Trees, parking functions, syzygies, and deformations of monomial ideals. Transactions of the American Mathematical Society, 356(8):3109–3142, 2004.
    .bib
    @article{PostnikovShapiro2004,
  author = {Alexander Postnikov and Boris Shapiro},
  title = {Trees, parking functions, syzygies, and deformations of monomial ideals},
  journal = {Transactions of the American Mathematical Society},
  volume = {356},
  number = {8},
  pages = {3109--3142},
  year = {2004},
  doi = {10.1090/S0002-9947-04-03547-0},
  eprint = {math/0301110},
  url = {https://doi.org/10.1090/S0002-9947-04-03547-0}
}
  15. [WW12]Jinkui Wan and Weiqiang Wang. Lectures on spin representation theory of symmetric groups. Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 7(1):91–164, 2012.
    .bib
    @article{WanWang2012Spin,
  author = {Jinkui Wan and Weiqiang Wang},
  title = {Lectures on spin representation theory of symmetric groups},
  journal = {Bulletin of the Institute of Mathematics, Academia Sinica (New Series)},
  volume = {7},
  number = {1},
  pages = {91--164},
  year = {2012},
  eprint = {1110.0263},
  url = {https://www.math.sinica.edu.tw/bulletins/20121/2012104.pdf}
}

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