Research Statement
Nicholas Ovenhouse
My research is mainly in the area of algebraic combinatorics, with strong connections to geometry and dynamical systems. Specically, I am interested in problems related to, and inspired by, the theory of cluster algebras, in particular those having to do with Poisson geometry and integrable systems. Below are summaries of some of my published results, along with some background. Possible directions of future research are also discussed.
Cluster Algebras
Cluster algebras are special kinds of commutative algebras. They are subrings of a fraction eld Q(x1, . . . , xn) whose generators are dened by a recursive combinatorial procedure called mutation [FZ02]. One of the earliest and most basic results in the theory is what is known as the Laurent Phenomenon. It says that each of the generators (called cluster variables), which are, by denition, rational functions, are in fact always Laurent polynomials. It is therefore a point of interest to explicitly describe, combinatorially, the terms that occur in these Laurent polynomial expressions.
One interesting class of cluster algebras come from two-dimensional triangulated surfaces [FST08]. The clus- ter variables represent coordinate functions on the decorated Teichmüller space of the surface. In this case several combinatorial interpretations of the Laurent polynomial expansions are known [Sch08] [Yur19] [MSW11].
The simplest case is that of a triangulated polygon. The cluster variables correspond to the diagonals of the polygon. For a given diagonal, Schi er proved that the terms in the Laurent expansion are indexed by combinatorial objects called “T -paths”, which are certain types of paths drawn on the triangulation, which begin and end at the same endpoints as the given diagonal [Sch08]. An example is pictured below for n = 5, showing all T -paths from vertex 2 to vertex 5. The edges of each T -path are colored alternately blue and red.
1 1 1
2 5 2 5 2 5
3 4 3 4 3 4 x23x15 x12x34x15 x12x45 x13 x13x14 x14
x23x15 x12x34x15 x12x45 x25 = + + x13 x13x14 x14
1 My Contributions
Recently, a super-manifold version of the decorated Teichmüller space was introduced by Penner and Zeitlin [PZ19], where the authors gave a super-version of the mutation rule. In this setup, there is a super-algebra with even gener- ators corresponding to the diagonals of a xed triangulation, and odd generators (labelled µi in the example below) corresponding to the triangles. In joint work with Gregg Musiker and Sylvester Zhang, we gave a generalization of Schi er’s T -path description of the Laurent expansion in this super case. More specically, we dened combina- torial objects called “super T -paths” on a triangulated polygon. They are like ordinary T -paths, but they also allow “teleportation” between non-adjacent triangles. Our main theorem is the following.
Theorem. [MOZ21] Let Tij denote the set of all super T -paths from i to j. For a super T -path α ∈ Tij, there is a corresponding monomial xα in the initial variables so that X xij = xα
α∈Tij
An example is shown below (generalizing the example above). Teleportations are shown with dotted lines.
x23x15 x12x34x15 x12x45 x13 x13x14 x14
x12x15 µ1µ2 x12x15 µ1µ3 x12x15 µ2µ3
x23x15 x12x34x15 x12x45 x25 = + + + x12x15 (µ1µ2 + µ1µ3 + µ2µ3) x13 x13x14 x14
We also discovered connections to super frieze patterns [MGOT15] and super cluster algebras [OS18]. In partic- ular, we proved the following.
Theorem. [MOZ21] The elements of the super-algebra coming from a triangulated polygon can be arranged into a super frieze pattern (in the sense of [MGOT15]). In particular, the super frieze relations are equivalent to the super Ptolemy (mutation) relation.
2 Future Plans
Super Teichmüller Spaces: I plan to continue to investigate (with my co-authors) the analogy between these super Laurent expansions and the classical case of ordinary cluster algebras. In particular, we would like to generalize our formulas to surfaces of higher genus, and possibly also to punctured surfaces. Higher Teichmüller Spaces: Joint work with Andrew Claussen was documented in Claussen’s thesis [Cla20], where we investigated the Laurent expansions of cluster variables in higher Teichmüller spaces corresponding to the group SL3 (the “ordinary” case described above corresponds to SL2), and gave a description in terms of yet another analogue of T -paths. We obtained our results only for special types of triangulated polygons. I would like to prove our results in full generality, and possibly extend to other surfaces of higher genus.
Poisson Geometry and Integrable Systems
An associative, commutative algebra A is called a Poisson algebra if it is also a Lie algebra (with Lie bracket denoted {−, −}) subject to the “Leibniz identity”, which relates the Lie and associative products:
{x, y · z} = {x, y}· z + y ·{x, z}
Usually Poisson algebras are encountered as algebras of functions on some manifold or algebraic variety, in which case the space is called a Poisson manifold or Poisson variety. Let T : M → M be a Poisson mapping on a Poisson manifold M. Repeated application of T may be considered as a dynamical system with discrete time. The map T is said to be “completely integrable” if there exist functions ∞ f1, . . . , fk ∈ C (M) (k is some number depending on the rank of the Poisson bracket) such that {fi, fj} = 0, and additionally each fi is invariant under T . My research focuses on integrable systems which can be interpreted through the lens of cluster algebras. In particular this means that the map T : M → M, describing the dynamical system, can be written as a composition of cluster mutations.
Integrable Systems on a Cylinder
Consider a network (weighted directed graph) embedded on a cylinder, with some source vertices on one boundary circle, and some sink vertices on the other. An example is pictured below:
d 1 1 ! dc dλ a c B(λ) = b + adc adλ 2 2 b
One may then consider the space of all choices of edge weights, which is just RN (where N is the number of edges). There is a relatively simple Poisson structure one can dene on this space [GSV12]. For two edges incident to a common vertex, their edge weights x and y have Poisson bracket {x, y} = xy or {x, y} = 0, depending on their relative position. To such networks one can associate a path-weight matrix B(λ), whose entry bij(λ) is a Laurent polynomial which is the sum of all path weights from source i to sink j, and powers of λ record the winding number around
3 the cylinder. It was proven in [GSV12] that the Poisson brackets between entries of this matrix satisfy the Sklyanin r-matrix bracket (for the trigonometric r-matrix associated to SLn). This is expressed in the formula:
{B(λ),B(µ)} = r(λ, µ),B(λ) ⊗ B(µ)
From this expression, one can show that the collection of spectral invariants (i.e. the coe cients of the polynomials tr(B(λ)k)) Poisson-commute.
My Contributions
In recent work with Semeon Arthamonov and Michael Shapiro, we generalized these results to the case where the edge weights on the graph have noncommutative weights (thought of either as formal variables or matrix-valued weights). In place of an ordinary (commutative) Poisson bracket, we used the notion of a double bracket [VdB08], which is a bilinear operation (denoted {{−, −}}) on a (noncommutative) algebra A, which takes values in A ⊗ A (rather than just A). For example, the bracket of two generators might look like {{x, y}} = x ⊗ y or {{x, y}} = 1 ⊗ xy. The double bracket is required to satisfy some relations that resemble the axioms of a Poisson algebra. Let µ: A ⊗ A → A be the multiplication map (i.e. µ(x ⊗ y) = xy). Composing the double bracket with µ, we obtain an operation {−, −} = µ◦{{−, −}} which more closely resemebles an ordinary Poisson bracket. Going further, we can dene an operation on the cyclic space A\ := A/[A, A] by taking an equivalence class after applying {−, −}. We denote the induced bracket on A\ by h−, −i. There are natural reasons for considering such induced brackets. They give actual (ordinary) Poisson brackets on the “representation spaces” Repn(A) = Hom(A, Matn×n)/GLn, which are the moduli spaces of n-dimension representations of A [CB11].
Our rst main result is a direct generalization of the r-matrix formula above. There is a free associative algebra whose generators label the edges in the network, and a double bracket which mimics the Poisson bracket in the commutative case. For two edges incident to a common vertex, with labels x and y, their double bracket is {{x, y}} = x ⊗ y or {{x, y}} = y ⊗ x or {{x, y}} = 0.
Theorem. [AOS20] Let B(λ) be the path-weight matrix of a network with noncommutative edge weights. Then the entries of B satisfy the relations {{B(λ),B(µ)}} = r(λ, µ),B(λ)L ⊗ B(µ)R τ
The notations on the right-hand side describe noncommutative versions of the expressions in the classical case. For a matrix M with entries mij ∈ A, the matrix ML has entries mij ⊗ 1 ∈ A ⊗ A, and similarly MR has entries τ 1 ⊗ mij. The notation [−, −]τ is the “twisted commutator” which means [a, b]τ = ab − (ba) , where τ means (x ⊗ y)τ = y ⊗ x.
We also proved that the spectral invariants of B(λ) Poisson-commute as in the commutative case, but here it is only true for the induced bracket h−, −i on the quotient A\ = A/[A, A].
j P i \ Theorem. [AOS20] Let tr B(λ) = i Hijλ . Then the images of the Hij’s commute under the bracket on A :
hHij,Hk`i = 0 (for all i, j, k, `)
4 Main Application: The Pentagram Map
The pentagram map [Sch92] is dened on the space of plane polygons as follows. Given a polygon in the plane, draw the diagonals connecting vertex i to vertex i + 2. The intersection points of these diagonals will give the vertices of a new polygon. The map sending the old polygon to the new one is called the pentagram map. It is pictured below:
More generally, we can consider the polygons in the projective plane P2.A twisted polygon in P2 is a bi-innite 2 sequence of points (pi)i∈Z in P together with a projective transformation M ∈ SL3 (called the monodromy) such 2 that pi+n = Mpi for each i. If we let P denote the moduli space of twisted polygons in P , then the pentagram map makes sense as a map on P. This map was proven to be completely integrable [OST10]. That is, there exists a maximal family of independent functions fi on P, which are invariant under the pentagram map, and a Poisson structure in which {fi, fj} = 0 for all i, j.
Recently, Marí-Be a and Felipe introduced a generalization of the pentagram map where the twisted polygons are in a Grassmannian (rather than projective space) [FMB19].
As an application of the techniques mentioned above, I interpreted the space of Grassmann polygons as the space of matrix-valued weights on some cylindrical network to obtain some statement of integrability of this system.
Theorem. [Ove20]
(a) The space of Grassmann polygons is isomorphic to the space of matrix-valued weights on a cylinderical network (b) The induced bracket h−, −i on A\ = A/[A, A] gives an actual Poisson bracket on the space of Grassmann polygons
j (c) Let Hij be as above (the coe cients of tr(B(λ) )). They are invariant under the Grassmannian pentagram map, and they Poisson-commute.
Future Plans
Technical Issues of Integrability: The theorem mentioned above gives a family of invariant functions which Poisson-commute. However, to technically satisfy the denition of complete integrability, it must be checked that su ciently many of these functions are algebraically independent. To do so would require computing the rank of the Poisson structure, and demonstrating that the correct number of these functions are independent. Generalizations of the Noncommutative Framework: Certain simplifying assumptions about the weighted networks were made in the results referenced above. In particular, the networks were assumed to be acyclic. It would be interesting to extend the results to networks which have directed cycles. There are also dimer models
5 which are closely related to the network description [GK13], and it would be nice to see in what sense the dimer model can also be generalized to the noncommutative setup. Connection with Refactorization Mappings: Recently, Izosimov gave a unied framework which describes many di erent higher-dimensional generalizations of the pentagram map. This framework is based on the refactorization of di erence operators. It is closely related to the methods described above, since in the network description, the pentagram map can be seen as a refactorization of the path-weight matrix B(λ). It would be interesting to investigate more carefully the connection between Izosimov’s explicit Poisson bracket formulas and the ones obtained from the noncommutative techniques described above.
References
[AOS20] S Arthamonov, N Ovenhouse, and M Shapiro. Noncommutative networks on a cylinder. arXiv preprint arXiv:2008.02889, 2020. [CB11] William Crawley-Boevey. Poisson structures on moduli spaces of representations. Journal of Algebra, 325(1):205–215, 2011.
[Cla20] Andrew Claussen. Expansion posets for polygon cluster algebras. arXiv preprint arXiv:2005.02083, 2020.
[FMB19] Raúl Felipe and Gloria Marí Be a. The pentagram map on grassmannians. In Annales de l’Institut Fourier, volume 69, pages 421–456, 2019. [FST08] Sergey Fomin, Michael Shapiro, and Dylan Thurston. Cluster algebras and triangulated surfaces. part i: Cluster complexes. Acta Mathematica, 201(1):83–146, 2008. [FZ02] Sergey Fomin and Andrei Zelevinsky. Cluster algebras i: foundations. Journal of the American Mathe- matical Society, 15(2):497–529, 2002. [GK13] Alexander B Goncharov and Richard Kenyon. Dimers and cluster integrable systems. In Annales scien- tiques de l’École Normale Supérieure, volume 46, pages 747–813, 2013. [GSV12] Michael Gekhtman, Michael Shapiro, and Alek Vainshtein. Poisson geometry of directed networks in an annulus. Journal of the European Mathematical Society, 14(2):541–570, 2012. [MGOT15] Sophie Morier-Genoud, Valentin Ovsienko, and Serge Tabachnikov. Introducing supersymmetric frieze patterns and linear di erence operators. Mathematische Zeitschrift, 281(3):1061–1087, 2015. [MOZ21] Gregg Musiker, Nicholas Ovenhouse, and Sylvester Zhang. An expansion formula for decorated super- teichmüller spaces. arXiv preprint arXiv:2102.09143, 2021.
[MSW11] Gregg Musiker, Ralf Schi er, and Lauren Williams. Positivity for cluster algebras from surfaces. Ad- vances in Mathematics, 227(6):2241–2308, 2011. [OS18] Valentin Ovsienko and Michael Shapiro. Cluster algebras with grassmann variables. arXiv preprint arXiv:1809.01860, 2018. [OST10] Valentin Ovsienko, Richard Schwartz, and Serge Tabachnikov. The pentagram map: a discrete integrable system. Communications in Mathematical Physics, 299(2):409–446, 2010. [Ove20] Nicholas Ovenhouse. Non-commutative integrability of the grassmann pentagram map. Advances in Mathematics, 373:107309, 2020.
6 [PZ19] RC Penner and Anton M Zeitlin. Decorated super-teichmüller space. Journal of Di erential Geometry, 111(3):527–566, 2019.
[Sch92] Richard Schwartz. The pentagram map. Experimental Mathematics, 1(1):71–81, 1992.
[Sch08] Ralf Schi er. A cluster expansion formula (an case). the electronic journal of combinatorics, 15(R64):1, 2008.
[VdB08] Michel Van den Bergh. Double poisson algebras. Transactions of the American Mathematical Society, 360(11):5711–5769, 2008.
[Yur19] Toshiya Yurikusa. Cluster expansion formulas in type a. Algebras and Representation Theory, 22(1):1–19, 2019.
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