FKA (PDF)

A Contemporary Review
of Fomin-Kirillov Algebras
J. Alex Ruble
North Carolina State University

Target Journal: European
Journal of Combinatorics
• Specializes in combinatorial problems in pure
mathematics
• Establishes links between combinatorics and other
fields (algebra, (co)homology, computing)
• Emphasizes rigor and enforces high refereeing
standards
• Has a five-year impact factor of .703

A Fomin-Kirillov algebra is a quadratic algebra
defined by three relations
• For n > 2, the FKA of order n is denoted ℰn
• ℰn has generators [ij] for 1 ≤ i < j ≤ n (with [ji] ≝ -[ij])1
Structure-defining relations1 (for i, j, k, l pairwise distinct)
1. [ij]2 = 0 (elements are nilpotent)
2. [ij][jk] + [jk][ki] + [ki][ij] = 0 (cycled 3-permutations sum to 0)
3. [ij][kl] – [kl][ij] = 0 (ℰn is in some sense braided)

Dimensionality
• ℰn‘s generator set grows as n2
• |ℰn|’s growth is dramatic
(qualitatively exponential,
quantitatively unknown)2,3
• Closed formulas for the Hilbert
series of ℰn (ℋn(q)) are known to
be the product of q-numbers for
n≤51
• ℋ6(q) cannot be written as the
sum of q-numbers1

Commutative Quotients
• Define ℰnab to be the subalgebra generated by stipulating4
[ij][kl] = [kl][ij]
• This imposes commutativity on the algebra, greatly simplifying its
internal structure
• ℰnab is much easier to analyze (via hyperplane arrangements)5
• |ℰnab| = n! 5
• ℋnab (q) = (1+q)(1+2q)…(1+(n-1)q) 5

Graph Subalgebras
• What if we restrict the generators to some certain proper subset of
C([n], 2)?
• For a graph G = (V, E), define ℰG, with generators [ij] exactly
corresponding to the edges of G 4
• We can define ℰGab in the same way as before
• The Orlik-Terao algebra for a hyperplane arrangement9 can be defined
for a graph4 (call it UG)
• (Thm. 14) ℰGab and UG are isomorphic (!)
• What does this mean?

Chromatic Polynomial
• (Cor.4) It means ℰGab has Hilbert series ℋGab(q) = (-q)nχG(-q-1), where
χG(q) is the chromatic polynomial of G
• This affords a straightforward analysis of ℰGab!
Hyperplane arrangement

Graph

Algebra

Coloring!

Where does research go from here?
• What is |ℰn|?
• What is the positive
span of ℰn (ℰn+)?
• Which Schubert
polynomial evaluations
are in ℰn+?
• How can we interpret
the Dunkl elements?

Dunkl
elements7
θi = [ij]

Σ
j≠i

Polygon
triangulations

Plane
partitions

Fuss-Catalan
numbers

References
1.

S. Fomin, A.N.Kirillov. Quadratic algebras, Dunkl elements, and Schubert calculus. Advances in Geometry, Progress in Mathematics, vol.
172 (1999)

2.

J. Blasiak, R.I. Liu, K. Mészáros. Subalgebras of the Fomin-Kirillov algebra. Preprint. arXiv:1310.4112 (2012)

3.

L. Vendramin. Fomin-Kirillov algebras. “Nichols algebras and Weyl groupoids.” Oberwolfach miniworkshop colloquium, arXiv:1210.5423
(2012)

4.

R.I. Liu. On the commutative quotient of Fomin-Kirillov algebras. European Journal of Combinatorics, vol. 54 (2016)

5.

I.M. Gelfand, A.N. Varchenko. Heaviside functions of a configuration of hyperplanes. Funct. Anal. Appl., vol. 21 (1988)

6.

P. Orlik, L. Solomon. Combinatorics and topology of complements of hyperplanes. Invent. Math., vol. 56 (1980)

7.

On some quadratic algebras I.5: Combinatorics of Dunkl and Gaudin elements, Schubert, Grothendieck, Fuss-Catalan, universal Tutte,
and reduced polynomials. SIGMA (2016)

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EJOC cover image: http://cdn.elsevier.com/cover_img/622824.gif
Generator/order plot: J. Alex Ruble, MATLAB
Hyperplane arrangement image: http://www.math.lsa.umich.edu/~jrs/gifs/A3s.gif
Graph image: https://upload.wikimedia.org/wikipedia/commons/thumb/5/5b/6n-graf.svg/220px-6n-graf.svg.png
Algebra image: http://ocw.mit.edu/courses/mathematics/18-705-commutative-algebra-fall-2008/18-705f08.jpg
Graph coloring image: https://upload.wikimedia.org/wikipedia/commons/thumb/9/90/Petersen_graph_3-coloring.svg/2000pxPetersen_graph_3-coloring.svg.png
Plane partition image: https://upload.wikimedia.org/wikipedia/commons/thumb/1/14/Partition3D.svg/220px-Partition3D.svg.png
Polygon triangulation image: https://upload.wikimedia.org/wikipedia/commons/thumb/c/c2/Triangulation_3-coloring.svg/2000pxTriangulation_3-coloring.svg.png
Dyck path image: https://upload.wikimedia.org/wikipedia/commons/thumb/0/01/Dyck_lattice_D4.svg/220px-Dyck_lattice_D4.svg.png