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.
Lectures:
Lecture 1 (Title TBD),
Lecture 2 (Title TBD),
Lecture 3 (Title TBD),
Lecture 4 (Title TBD)
Exercises: to be added later.
Exercise session leaders: Hsin-Chieh Liao, Anastasia Nathanson
Some related talks
Background reading
- Koszulity orignal sources, books, and surveys
- Priddy, Koszul resolutions
- Fröberg, Determination of a class of Poincare series
- Fröberg, Koszul algebras (survey)
- Backelin and Fröberg, Koszul algebras, Veronese subrings and rings with linear resolutions
- Polishchuk and Positselski's book, "Quadratic algebras" (AMS University Lecture Series)
- McCullough and Peeva, Infinite graded free resolutions
- Jöllenbeck and Welker, Resolution of the residue class field via algebraic discrete Morse theory
- Ufnarovskii, V.A., Combinatorial and asymptotic methods in algebra
- Recent and not-so-recent papers
- Almousa, Reiner and Sundaram, Koszulity, supersolvability, and Stirling representation
- Angarone, Nathanson and Reiner, Chow rings of matroids as permutation representations
- Backman and Liu, A regular unimodular triangulation of the matroid base polytope
- Bruns, Herzog and Vetter, Syzygies and walks
- Coron,
Supersolvability of built lattices and Koszulness of generalized Chow rings
- Dorpalen-Barry, The Varchenko-Gelfand ring of a cone
- Dotsenko, Homotopy invariants for M_{0,n} via Koszul duality
- Eisenbud, Reeves and Totaro, Initial ideals, Veronese rings, and rates of algebras
- Faber, Juhnke-Kubitzke, Lindo, Miller, R. G., Seceleanu, Canonical resolutions over Koszul algebras
- Jambu, Koszul algebras and hyperplane arrangements
- Josuat-Verges and Nadeau, Koszulity of dual braid monoid algebras via cluster complexes
- Maestroni and McCullough, Chow rings of matroids are Koszul
- Peeva, Hyperplane arrangements and linear strands in resolutions
- Peeva, Reiner and Sturmfels, How to shell a monoid
- Reiner and Stamate, Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals
- Reiner and Welker, On the Charney-Davis and Neggers-Stanley Conjectues
- Roos, Some non-Koszul algebras
- Shelton and Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements
- VandeBogert, Ribbon Schur functors
Long-standing open problems and questions
- 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.
- 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).
- 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.
- 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?
- 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.
- Thrall's Problem: give a combinatorial formula for the irreducible expansion of the higher Lie characters, essentially describing the type A reflection arrangement's graded Varchenko-Gelfand ring and its Orlik-Solomon algebra. See Schocker.
Not as long-standing open problems and questions
- Is the S_n-representation on the cohomology of M0,n a permutation representation?
See Ramadas-Silversmith and Dotsenko.
- Conjectures 4.7, 4.11, 4.23 on matroid Chow rings
from our paper with Angarone and Nathanson
- 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
- 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
- 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?