MPI Leipzig Summer School 2024 in Algebraic Combinatorics

Vic Reiner: The Koszul property in algebraic combinatorics

Many graded algebras appearing in Algebraic Combinatorics turn out to have the Koszul Property, which is defined algebraically. This property has interesting combinatorial consequences for the Hilbert series of the algebra, and naturally leads one to study its partner, called its Koszul dual algebra. Examples of Koszul algebras include polynomial algebras, exterior algebras, Stanley-Reisner rings of posets and flag complexes, Orlik-Solomon algebras for supersolvable hyperplane arrangements, Veronese and Segre rings, Hibi rings, Chow rings of matroids, and partial commutation monoid algebras. In addition to these examples, this course will discuss basic theory of Koszul algebras: their combinatorial properties, constructions, and their interaction with topics such as affine semigroup rings, walks in digraphs, representation stability, unimodality, log-concavity, the Polya frequency property and the Charney-Davis-Gal conjecture.
Lecture 1: Motivation, definition of Koszul algebras
Lecture 2: Methods for proving Koszulity, and more examples
Lecture 3: Bar complex, topology and inequalities
Lecture 4: Group actions

Exercises for Session 1
Exercises for Session 2
Exercise session leaders: Hsin-Chieh Liao, Anastasia Nathanson

Some related talks

Background reading

Long-standing open problems and questions
  1. Is Neil White's matroid basis ring a quadratic algebra, and is it Koszul? See White, Blasiak, Kashiwabara, Lason-Michalek
    What about for polymatroids? See Herzog-Hibi.
  2. The Charney-Davis-Gal Conjecture: gamma-nonnegativity for flag simplicial homology spheres, or equivalently, Koszul Gorenstein Stanley-Reisner rings.
    See this paper and note by Gal, and Athanasiadis.
    See also a stronger conjecture by
    Nevo-Petersen with results by Aisbett-Volodin.
    Lastly, note the cautionary tale in
    D'Ali-Venturello (Cor. 7.4).
  3. For the Orlik-Solomon algebra OS(M) of a matroid M, does OS(M) Koszul imply M is supersolvable? See Yuzvinsky, Peeva.
    Same for the (graded) Varchenko-Gelfand ring VG(M) of an oriented matroid M: Does VG(M) Koszul imply M is supersolvable? See Dorpalen-Barry.
  4. For which matroids M is the quadratic closure of their Orlik-Solomon algebra O(M) a Koszul algebra? See Peeva
    Does this hold for the matroid of the reflection arrangement of type D_n?
  5. Bøgvad's Conjecture: the toric ring of a smooth projectively normal toric variety is Koszul; see Bøgvad
  6. For which quadratic monomial quotients of the free associative algebra is the Hilbert series a Polya-Frequency sequence? See Chapter 7 of Brenti's thesis.
  7. Thrall's Problem: give a combinatorial formula for the irreducible expansion of the higher Lie characters of Sn,
    essentially describing the type A reflection arrangement's graded Varchenko-Gelfand ring and its Orlik-Solomon algebra. See
Not as long-standing open problems and questions
  1. Is the Sn-representation on the cohomology of M0,n a permutation representation?
    See Ramadas-Silversmith and Dotsenko.
  2. Let A=S/I where S=k[x1,...,xn] is the commutative polynomial algebra, and assume A is a Koszul algebra.
    Do the Betti numbers bi=biS(A) in the finite minimal free resolution of A as an S-module satisfy bi ≤ ( b1 choose i )?
    See Conca, De Negri, Rossi.
  3. Conjectures 4.7, 4.11, 4.23 on matroid Chow rings
    from our paper with Angarone and Nathanson
  4. Problems and conjectures on Koszul duals of type A Orlik-Solomon algebras and graded Varchenko-Gelfand rings
    from Sections 10, 11, 12 of our paper with Almousa and Sundaram
  5. For a poset P, is the ideal of P-partitions within the ring of weak P-partitions always a nongraded Koszul module?
    See Sections 8,9,12 of our paper with Féray
  6. For which finite subgroups of GLn(k) acting on a polynomial ring S=k[x1,...,xn] does one have ...
    • ... that the invariant ring SG is a nongraded Koszul algebra?
    • ... the polynomial ring S is a nongraded Koszul module over the invariant ring SG?