Cluster complexes and generalized associahedra

Christian Stump, Universität Hannover, Feb 8, 2012

1. Cluster Algebras

A cluster algebra $\mathcal{A}$ of rank $n$ is a subring of the ring of rational functions $\mathbb{Q}(x_1,\ldots,x_n)$ equipped with

• a distinguished set of generators (cluster variables),
• grouped into overlapping subsets (clusters) of cardinality $n$.

The main examples of cluster algebras are coordinate rings of important algebraic varieties such as

• homogeneous coordinate rings of Grassmannians and of Schubert varieties.

Starting with an initial cluster $C = \{x_1, \ldots, x_n\}$, there is for any $1 \leq k \leq n$ a combinatorial rule in form of

• a skew-symmetrizable $(n \times n)$-integer matrix $M = (m_{ij})$,
  
                  or equivalently
  
  
• a quiver on $n$ vertices with edge labels $(m_{ij}, -m_{ji})$.

to construct another cluster $\mu_k(C) = C - \{x_k\} \cup \{\tilde x_k\}$, and a new skew-symmetrizable matrix $\mu_k(M) = \widetilde M$.

• The pair $(C,M)$ is called initial seed,
• the pair $\mu_k(C,M) = (\mu_k(C),\mu_k(M))$ is obtained from $(C,M)$ by mutating in direction $k$,
• the mutation $\mu_k$ is an involution, $\mu_k^2 = \operatorname{id}$.
  

As a first example, we consider

Cluster Algebras of rank 2

For two positive integers $b$ and $c$, define an algebra $\mathcal{A}(b,c)$

• the cluster variables are the elements $x_m$ for $m \in \mathbb{Z}$, defined recursively the the exchange relation

$$x_{m-1} x_{m+1} = \begin{cases} x_m^b & \text{if m is odd} \\ x_m^a & \text{if m is even} \end{cases}$$

• iterating this relations, one can express each $x_m$ as a rational function in $x_1, x_2$.
• The clusters are $\{ x_m, x_{m+1} \}$, and we can reach each cluster by a series of exchanges.

Defining the mutation involution $\mu_k$

$$x_k \mu_k(x_k) = \prod x_i^{[b_{ik}]_+} + \prod x_i^{[-b_{ik}]_+}$$

$$\tilde b_{ij} = \begin{cases} -b_{ij} & \text{ if } i = k \text{ or } j = k \\ b_{ij} + [b_{ik}]_+ [b_{kj}]_+ - [-b_{ik}]_+ [-b_{kj}]_+ & \text{ otherwise } \end{cases}$$

Theorem (Laurent phenomenon, Fomin-Zelevinsky)

All cluster variables are indeed Laurent polynomials in $x_1,\ldots,x_n$.

Theorem (Finite type cluster algebras, Fomin-Zelevinsky)

Let $\mathcal{A}$ be a cluster algebra. Then

$\mathcal{A}$ has a finite number of cluster variables

if and only if

Some mutation matrix for $\mathcal{A}$ is a Cartan matrix of finite type.

Connections of Cluster Algebras to other fields

In the past ten years, cluster algebras have been found to be related to a number of other topics such as

• quiver representations,
• tropical geometry,
• canonical bases of semisimple algebraic groups,  
• total positivity,
• generalized associahedra,
• Poisson geometry, and
• Teichmüller theory.

In the past 12 months, there appeared

• about 35 papers on the arXiv having "cluster algebra" in the title,
• and about 100 papers having it in the abstract.

2. The cluster complex

Let $\mathcal{A}$ be a cluster algebra of finite type. The cluster complex $C(\mathcal{A}_W)$ is the simplicial complex with

• vertices being cluster variables,
• facets being clusters.

The main aim of this talk is a presentation of

• a new combinatorial description of the cluster complex in finite types
  (joint work with C. Ceballos and J.-P. Labbé)
  
  
• a construction of a polytope which graph is exactly the exchange graph of the cluster algebra / cluster complex
  (joint work with V. Pilaud)

Both constructions together provide a completely new approach to cluster complexes in finite types.

4. Subword complexes

(introduced by A. Knutson and E. Miller 2003)

Let

• $(W,S)$ be a Coxeter system,
• $Q$ be a (not necessarily reduced)  word in $S$, and
• $w$ be an element in $W$.

The subword complex $\Delta(Q,w)$ is the simplicial complex with

• vertices being (positions of) letters in $Q$,
• facets being complements of subwords of $Q$ that are reduced expressions for $w$.
  

Theorem (Ceballos-Labbé-S.)

The cluster complex can be obtained as a subword complex for a well-chosen word $Q = {\bf cw}_\circ$ and the longest element $w_\circ \in W$,

$$C(\mathcal{A}_W) \cong \Delta(Q,w_\circ).$$

5. Generalized associahedra

Theorem (Pilaud-S.)

• The brick polytope $\mathcal{B}({\bf cw}_\circ)$ realizes the subword complex $S({\bf cw}_\circ)$ and thus the cluster complex $C(\mathcal{A}_W)$.
• The above realization is an affine translation of the polytopal realizations of Hohlweg-Lange-Thomas and of Chapoton-Fomin-Zelevinsky.

Corollary

The brick polytope $\mathcal{B}({\bf cw}_\circ)$

• gives an explicit way to realize generalized associahedra,
• gives a vertex and a facet description of the generalized associahedra,
• gives a natural Minkowski decomposition of generalized associahedra into Coxeter matroid polytopes.