Combinatorics and Representations of
0-Hecke Algebras

Instructor: Vassilis Moustakas
Email: vd.moustakas AT gmail.com
Office: 119 
Office hours: By appointment (or just pass by my office)
Class Details: Tuesday & Thursday (for exact details see below)

Course Description

We will study the combinatorics and representation theory of 0-Hecke Algebras. Hecke algebras are deformations of the group algebra of the symmetric group, whose representation theory connects those of the symmetric group and the quantum groups. Hecke algebras associated with general Coxeter groups appear in diverse areas such as harmonic analysis, quantum groups, knot theory, algebraic combinatorics and statistical physics. The representation theory of the symmetric group is closely connected to the algebra of symmetric functions, Sym, through the so-called Frobenius characteristic map. Sym admits two notable generalizations: the algebra of quasisymmetric functions, QSym, and the algebra of noncommutative symmetric functions, NSym. We will explore the relationship between these algebras and representations of 0-Hecke algebras through a quasisymmetric characteristic map. These combinatorial Hopf algebras play a central role in contemporary algebraic combinatorics, aspects of which we will discuss throughout the course.   

Course Contents

• Representations of finite-dimensional algebras

• Combinatorial Hopf algebras

• Representations of finite groups 

• Algebras of symmetric, quasisymmetric and noncommutative symmetric functions

• The weak and Bruhat order on the symmetric group

• Iwahori-Hecke algebra of the symmetric group

• Representations of the 0-Hecke algebra

• 0-Hecke actions on posets, polynomials and tableaux (if time permits)

Prerequisites

Familiarity with linear and abstract algebra. Some knowledge of basic group representation theory would be helpful but is not strictly necessary. There are no combinatorial prerequisites for this course. 

Textbooks and other useful material

The following lists are by no means exhaustive and are meant as a guide to the subjects addressed. If you don't have access to any of the following material and want to, please contact me.

Book treatments on Iwahori–Hecke and 0-Hecke algebras include:

• Meliot - Representation Theory of Symmetric Groups

• Mathas - Iwahori–Hecke Algebras and Schur Algebras of the Symmetric Group

• Geck, Pfeiffer - Characters of Finite Coxeter Groups and Iwahori–Hecke Algebras

• Humphreys - Reflection Groups and Coxeter Groups

Books on the representation theory of algebras:

• Etingof et al. - Introduction to Representation Theory

• Assem, Simson, Skowronski - Elements of the representation theory of associative algebras

• Webb - A Course in Finite Group Representation Theory 

Books on the theory of symmetric functions, their generalizations and its connection to representation theory:

• Sagan - The Symmetric Group

• Stanley - Enumerative Combinatorics, Volume 2 (Chapter 7)

• Grinberg, Reiner - Hopf algebras in combinatorics

• Luoto, Mykytiuk, Van Willigenburg - An Introduction Quasisymmetric Schur Functions

• Hazewinkel, Gubareni, Kirichenko - Algebras, Rings and Modules: Lie Algebras and Hopf Algebras 

Other material include:

• Marberg's 2017 course on Coxeter Systems and Iwahori–Hecke Algebras

• Varizani's 2019 lectures on Hecke Algebras and Representation Theory (Part 1, Part 2, Part 3 and Part 4)

• Ardila's wonderful courses on Coxeter groups and Hopf Algebras and Combinatorics 

Projects

1. 0-Hecke–Clifford algebras and peak quasisymmetric functions

   • Bergeron, Hivert, Thibon - The peak algebra and the Hecke–Clifford algebras at q=0

   • Li - Representation theory of 0-Hecke–Clifford algebras

   • Petersen - Enriched P-partitions and peak algebras

2. Temperley–Lieb algebras and quasisymmetric functions

   • Hivert - Hecke algebras, difference operators, and quasi-symmetric Functions

   • Bergeron, Gagnon - Theexcedance quotient of the Bruhat order,  quasisymmetric varieties, and Temperley–Lieb algebras

3. Solomon descent algebras and quasisymmetric functions

   • Malvenuto, Reutenauer - Duality between quasi-symmetrical functions and the Solomon descent algebra

   • Garsia, Reutenauer - A decomposition of Solomon's descent algebra

4. 0-Ariki–Koike–Shoji algebras, Mantaci–Reutenauer descent algebras and colored quasisymmetric functions

   • Hivert, Novelli, Thibon - Yang–Baxter bases of 0-Hecke algebras and representation theory of 0-Ariki–Koike–Shoji algebras

   • Mantaci, Reutenauer - A generalization of Solomon's descent algebra for hyperoctahedral groups and wreath products

   • Baumann, Hohlweg - A Solomon descent theory for the wreath products G≀Sn

5. Quasisymmetric functions of matroids

   • Billera, Jia, Reiner - A quasisymmetric function for matroids

6. Quasisymmetric functions of bruhat intervals and Kazhdan–Lusztig polynomials

   • Billera, Brenti - Quasisymmetric functions and Kazhdan-Lusztig polynomials

   • Bjorner, Brenti - Combinatorics of Coxeter groups (Chapter 5)

7. Quasisymmetric Schur functions and 0-Hecke modules

   • Tewari, van Willigenburg - Modules of the 0-Hecke algebra and quasisymmetric Schur functions

   • Luoto, Mykytiuk, Van Willigenburg - An Introduction Quasisymmetric Schur Functions (Chapter 5)

8. Dual immaculate quasisymmetric functions and 0-Hecke modules

   • Berg, Bergeron, Saliola, Serrano, Zabrocki - Indecomposable modules for the dual immaculate basis of quasisymmetric functions

9. Hecke algebras of type A at roots of unity

   • Lascoux, Leclerc, Thibon - Hecke algebras at roots of unity and crystal bases of quantum affine algebras

10. Representations of quantum groups and the Jimbo–Schur–Weyl duality

    • Meliot - Representation Theory of Symmetric Groups (Chapter 5)

11. 0-Hecke algebra actions on coinvariants and ordered set partitions

    • Huang - 0-Hecke algebra actions on coinvariants and flags

    • Huang, Rhoades -  Ordered set partitions and the 0-Hecke algebra

Lectures

■ Thursday, February 20 • Saletta Riunioni • 10:00-11:00 • Organizational meeting

1. Tuesday, March 11 • Sala Riunioni • 11:00-13:00 • First lecture • notes

   • Coxeter–Moore presentation of the symmetric group

   • Matsumoto's theorem

   • Iwahori–Hecke algebras of type A

   • Notions: permutation, simple reflection, length, reduced expression, braid group, inversion

2. Thursday, March 13 • Aula Seminari • 11:00-13:00 • notes

   • Wedderburn–Artin theorem

   • Jordan–Hölder theorem

   • Krull–Schmidt theorem

   • Notions: algebra, ideal, representation, module; simple, semisimple, indecomposable, nilpotent element, Jacobson radical, composition series and factors, indecomposable projective, projective cover

3. Tuesday, March 18 • Sala Riunioni • 11:00-13:00 • notes

   • Notions: algebra, coalgebra, bialgebra, Hopf algebra, antipode, structure constants, graded dual

4. Thursday, March 20 • Sala Riunioni • 11:00-13:00 • notes

   • Maschke's theorem

   • Frobenius' reciprocity law

   • Notions: group representation, character, permutation representation, induction, restriction

5. Tuesday, March 25 • Sala Riunioni • 11:00-13:00 • notes

   • Hopf algebra of symmetric functions

   • Notions: monomial, elementary, complete homogeneous, power sum and Schur symmetric function, integer partition, (semi)standard Young tableau, Kostka number, dominance order

6. Thursday, March 27 • Aula Seminari • 11:00-13:00 • notes

   • Comultiplication of symmetric functions

   • Cauchy identity

   • Hall inner product

   • Littlewood–Richardson rule

   • Murnaghan–Nakayama rule

   • Frobenius characteristic map

   • Notions: skew partition, cycle type, Young subgroup, ribbon, ribbon tableau

7. Tuesday, April 1 • Sala Riunioni • 11:00-13:00 • notes

   • Hopf algebra of quasisymmetric functions

   • Notions: monomial and fundamental quasisymmetric function, composition, descent, shuffle, near-concatenation

8. Thursday, April 3 • Sala Riunioni • 11:00-13:00 • notes 

   • Specializations of quasisymmetric functions and permutation statistics

   • Hopf algebra of noncommutative symmetric functions

   • Notions: ribbon Schur function, principal specialization, major index, noncommutative complete homogeneous and ribbon symmetric function, forgetful map

9. Tuesday, April 8 • Sala Riunioni • 11:00-13:00 • notes

   • Hopf algebra of free quasisymmetric functions

   • The Weak order

   • Notions: standardization,  free quasiribbon function, free ribbon function, ascent, lattice, 

10. Thursday, April 10 • Aula Seminari • 09:00-11:00 • notes

    • Bruhat order

    • Iwahori–Hecke algebras of type A

    • Guest Lecture by Leonardo Patimo on Hecke algebras and Kazhdan–Lusztig representations

11. Tuesday, April 15 • Sala Riunioni • 11:00-13:00

    • The Jacobson radical of the 0-Hecke algebra

    • Simple modules over the 0-Hecke algebra

12. Thursday, April 17 • Aula Seminari • 15:00-17:00

    • Projective indecomposable modules over the 0-Hecke algebra

    • The quasisymmetric and noncommutative characteristic maps

13. Friday, May 2 • Sala Riunioni • 15:00-17:00

    • Cartan matrix of the 0-Hecke algebra

    • Crash course on the (combinatorial) theory of posets

    • Notions: poset, interval, order ideal, chain, rank function, graded poset, rank-generating function, lattice, distributive lattice, Birkhoff lattice, order polynomial, linear extension

    • Examples: chain poset, Boolean, partition, Young's and the subspace lattices.

14. Tuesday, May 6 • Sala Riunioni • 11:00-13:00 

    • Hall's theorem

    • Möbius inversion

    • Möbius function of the Bruhat order

    • Fundamental lemma of (P,ω)-partitions

    • 0-Hecke algebra poset modules

    • Notions: Möbius function, order complex, (reduced) Euler characteristic, f-vector, incidence (co)algebra, (P,ω)-partitions, Schur labelling, disjoint union of posets

15. Thursday May 8 • Sala Riunioni • 09:00-11:00 • Final lecture

    • 0-Hecke algebra poset modules

    • Notions: flag f-vector, flag h-vector, rank-selected subposet, edge labelling, R-labelling, flag quasisymmetric function, (P,ω)-Eulerian polynomial, chain polynomial, EL-labelling, Sn-EL-labelling

■ Tuesday, May 20 • Sala Riunioni • 09:00-13:00 • Presentations