The 0-Hecke algebra

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#The 0-Hecke algebra

The 0-Hecke algebra $H_n(0)$ in type $A$ is the $\setC$-algebra generated by $T_i,$ $i=1,\dotsc,n-1,$ subject to \[T_i^2 = T_i, \qquad T_iT_{i+1}T_i = T_{i+1}T_iT_{i+1}, \qquad T_jT_i = T_iT_j \text{ whenever } |i-j|\geq 2.\] The braid relations are the same as for the symmetric group, while the quadratic relation is idempotent rather than involutive. If $w=s_{i_1}\dotsm s_{i_\ell}$ is a reduced word, then \[T_w\coloneqq T_{i_1}\dotsm T_{i_\ell}\] is independent of the chosen reduced word. Thus $\{T_w:w\in\symS_n\}$ is a basis, and $\dim H_n(0)=n!.$

Example

For $n=3,$ the algebra $H_3(0)$ has basis \[T_e,\quad T_1,\quad T_2,\quad T_1T_2,\quad T_2T_1,\quad T_1T_2T_1.\] The braid relation identifies $T_1T_2T_1$ with $T_2T_1T_2,$ just as in the symmetric group.

#Representation theory

The representation theory of $H_n(0)$ parallels the representation theory of $\symS_n,$ but with compositions replacing partitions [KT97]. The simple $H_n(0)$-modules are one-dimensional and are indexed by compositions $\alpha\vDash n,$ or equivalently by subsets of $[n-1].$ We denote them by $F^\alpha.$

$ \text{Object} $ $ \text{Group algebra }\setC[\symS_n] $ $ \text{0-Hecke algebra }H_n(0)$ $ \text{Indexing set for simples} $ $ \text{partitions of }n $ $ \text{compositions of }n$ $ \text{Characteristic target} $ $ \spaceSym $ $ \spaceQSym$ $ \text{Simple/module basis} $ $ \schurS_\lambda $ $ \gessel_\alpha$ $ \text{Projective dual side} $ $ \spaceSym $ $ \mathrm{NSym}$

The quasisymmetric characteristic is the analogue of the Frobenius characteristic. It sends the simple module $F^\alpha$ to the fundamental quasisymmetric function $\gessel_\alpha.$

Example

For $n=3,$ the compositions and descent sets are \[(3)\leftrightarrow\varnothing,\qquad (1,2)\leftrightarrow\{1\},\qquad (2,1)\leftrightarrow\{2\},\qquad (1,1,1)\leftrightarrow\{1,2\}.\] On the simple module $F^{(1,2)},$ the generator $T_1$ acts by $0$ and $T_2$ acts by $1.$ Its quasisymmetric characteristic is $\gessel_{(1,2)}.$ This is the 0-Hecke analogue of sending an irreducible $\symS_n$-module to its Frobenius characteristic.

#QSym and NSym

The Grothendieck group of finite-dimensional $H_n(0)$-modules, summed over all $n,$ is identified with $\spaceQSym.$ The Grothendieck group of projective $0$-Hecke modules is identified with NSym. Under induction and restriction, these Grothendieck groups recover the dual Hopf-algebra structures on QSym and NSym [KT97].

This gives a representation-theoretic explanation for the duality between QSym and NSym. The fundamental quasisymmetric basis comes from simple modules, while the dual noncommutative side comes from projectives. More precisely, indecomposable projectives correspond to the noncommutative ribbon basis, which is dual to the fundamental quasisymmetric basis. Induction and restriction of 0-Hecke modules give the product and coproduct on the corresponding Hopf algebras.

#Schur-like bases

Many quasisymmetric Schur-like bases have 0-Hecke interpretations. For example, V. Tewari and S. v. Willigenburg show that quasisymmetric Schur functions arise as quasisymmetric characteristics of certain $H_n(0)$-modules [TW15]. Similar representation-theoretic models appear for dual immaculate, extended Schur, and related bases on the quasisymmetric Schur page.

Divided-difference and Demazure-type operators also have 0-Hecke flavor. For example, the operators used to construct key polynomials and non-symmetric Macdonald polynomials satisfy braid and idempotent-type relations closely related to $H_n(0).$

#Other types

The type $B$ 0-Hecke algebra replaces $\symS_n$ by the hyperoctahedral group. Y.-H. Kim and D. Searles construct type $B$ poset modules whose quasisymmetric characteristics give a representation-theoretic interpretation of type $B$ $P$-partition enumerators [KS26].

This is one instance of a broader pattern: once a Coxeter group has a Coxeter presentation, one can form a 0-Hecke algebra by replacing $s_i^2=1$ with an idempotent relation.

Bibliography

[KS26]Young-Hun Kim and Dominic Searles. Poset modules of the $0$-Hecke algebras of type ${B}$. arXiv:2601.22926, 2026.

.bib

@article{KimSearles2026x,
  author = {Young-Hun Kim and Dominic Searles},
  title = {Poset modules of the $0$-{H}ecke algebras of type ${B}$},
  year = {2026},
  eprint = {2601.22926},
  url = {https://arxiv.org/abs/2601.22926},
  journal = {arXiv e-prints}
}

[KT97]Daniel Krob and Jean-Yves Thibon. Noncommutative symmetric functions IV : Quantum linear groups and Hecke algebras at $q=0$. Journal of Algebraic Combinatorics, 6(4):339–376, 1997.

.bib

@article{KrobThibon1997,
  doi = {10.1023/a:1008673127310},
  url2 = {https://doi.org/10.1023/a:1008673127310},
  year  = {1997},
  publisher = {Springer Nature},
  volume = {6},
  number = {4},
  pages = {339--376},
  author = {Daniel Krob and Jean-Yves Thibon},
  title = {Noncommutative symmetric functions {IV} : Quantum linear groups and {H}ecke algebras at $q=0$},
  journal = {Journal of Algebraic Combinatorics}
}

[TW15]Vasu V. Tewari and Stephanie Willigenburg. Modules of the 0-Hecke algebra and quasisymmetric Schur functions. Advances in Mathematics, 285:1025–1065, November 2015.

.bib

@article{TewariWilligenburg2015,
  doi = {10.1016/j.aim.2015.08.012},
  url2 = {https://doi.org/10.1016/j.aim.2015.08.012},
  year  = {2015},
  month = nov,
  publisher = {Elsevier {BV}},
  volume = {285},
  pages = {1025--1065},
  author = {Vasu V. Tewari and Stephanie {van Willigenburg}},
  title = {Modules of the 0-{H}ecke algebra and quasisymmetric {S}chur functions},
  journal = {Advances in Mathematics}
}