Cluster complexes and generalized associahedra
system:sage


<h1 style="padding-left: 30px;">Cluster complexes and generalized associahedra</h1>
<p style="padding-left: 60px;"><span style="font-size: large;">Christian Stump, Universit&auml;t Hannover, Feb 8, 2012</span></p>
<h2>1. Cluster Algebras</h2>
<address><span style="font-size: small;">(introduced by S. Fomin and A. Zelevinsky 2001)</span></address>
<p>A <strong>cluster algebra</strong> $\mathcal{A}$ of <em><strong>rank</strong></em> $n$ is a subring of the <em>ring of rational functions</em> $\mathbb{Q}(x_1,\ldots,x_n)$ equipped with</p>
<ul>
<li>a distinguished set of generators (<em><strong>cluster variables</strong></em>),</li>
<li>grouped into overlapping subsets (<em><strong>clusters</strong></em>) of cardinality $n$.</li>
</ul>
<p>The <strong>main examples</strong> of cluster algebras are <strong>coordinate rings</strong> of important <strong>algebraic varieties</strong> such as</p>
<ul>
<li>homogeneous coordinate rings of Grassmannians and of Schubert varieties.</li>
</ul>
<p>Starting with an <em><strong>initial cluster</strong></em> $C = \{x_1, \ldots, x_n\}$, there is for any $1 \leq k \leq n$ a <em>combinatorial rule</em> in form of</p>
<ul>
<li>a <strong>skew-symmetrizable</strong> $(n \times n)$<strong>-integer matrix</strong> $M = (m_{ij})$,<br /><br />&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; or equivalently<br /><br /></li>
<li>a <strong>quiver</strong> on $n$ vertices with edge labels $(m_{ij}, -m_{ji})$.</li>
</ul>
<p>to construct another cluster $\mu_k(C) = C - \{x_k\} \cup \{\tilde x_k\}$, and a new skew-symmetrizable matrix $\mu_k(M) = \widetilde M$.</p>
<ul>
<li>The pair $(C,M)$ is called <strong><em>initial seed</em></strong>,</li>
<li>the pair $\mu_k(C,M) = (\mu_k(C),\mu_k(M))$ is obtained from $(C,M)$ by <em><strong>mutating in direction</strong></em> $k$<em><strong>,</strong></em></li>
<li>the mutation $\mu_k$ is an involution, $\mu_k^2 = \operatorname{id}$.<strong><em><br /></em></strong></li>
</ul>
<p>As a first example, we consider</p>
<p><strong>Cluster Algebras of rank 2</strong></p>
<p>For two positive integers $b$ and $c$, define an algebra $\mathcal{A}(b,c)$</p>
<ul>
<li>the cluster variables are the elements $x_m$ for $m \in \mathbb{Z}$, defined recursively the the <em><strong>exchange relation</strong></em></li>
</ul>
<p>$$x_{m-1} x_{m+1} = \begin{cases} x_m^b &amp; \text{if m is odd} \\ x_m^a &amp; \text{if m is even} \end{cases}$$</p>
<ul>
<li>iterating this relations, one can express each $x_m$ as a rational function in $x_1, x_2$.</li>
<li>The clusters are $\{ x_m, x_{m+1} \}$, and we can reach each cluster by a series of exchanges.</li>
</ul>

{{{id=2|
S = ClusterSeed(['R2',(1,1),2])
S.interact()
///
}}}

{{{id=23|
S = ClusterSeed(['R2',(1,1),2])
S.mutation_sequence([0,1,0,1,0],show_sequence=True)
///
<html><font color='black'><img src='cell://sage0.png'></font></html>
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}}}

<p><strong>Defining the mutation involution</strong> $\mu_k$</p>
<p>$$x_k \mu_k(x_k) = \prod x_i^{[b_{ik}]_+} + \prod x_i^{[-b_{ik}]_+}$$</p>
<p>$$\tilde b_{ij} = \begin{cases} -b_{ij} &amp; \text{ if } i = k \text{ or } j = k \\ b_{ij} + [b_{ik}]_+ [b_{kj}]_+ - [-b_{ik}]_+ [-b_{kj}]_+ &amp; \text{ otherwise } \end{cases}$$</p>
<p>&nbsp;</p>
<p><strong><span style="font-size: x-large;">Theorem</span> (Laurent phenomenon, Fomin-Zelevinsky)</strong></p>
<p>All cluster variables are indeed <em>Laurent polynomials</em> in $x_1,\ldots,x_n$.</p>
<p>&nbsp;</p>
<p><span style="font-size: x-large;"><strong>Theorem</strong></span><strong> (Finite type cluster algebras, Fomin-Zelevinsky)</strong></p>
<p>Let $\mathcal{A}$ be a cluster algebra. Then</p>
<p>$\mathcal{A}$ has a<strong> finite number of cluster variables</strong></p>
<p style="padding-left: 60px;">if and only if</p>
<p>Some mutation matrix for $\mathcal{A}$ is a <strong>Cartan matrix</strong> of finite type.</p>

{{{id=5|
S = ClusterSeed(['R2',(1,2),2])
S.interact()
///
}}}

{{{id=6|
S = ClusterSeed(['R2',(1,3),2])
S.interact()
///
}}}

{{{id=7|
S = ClusterSeed(['R2',(1,4),2])
S.interact()
///
}}}

<p><strong>Connections of Cluster Algebras to other fields</strong></p>
<p>In the past ten years, cluster algebras have been found to be related to a number of other topics such as</p>
<ul>
<li>quiver representations,</li>
<li>tropical geometry,</li>
<li>canonical bases of semisimple algebraic groups,&nbsp;&nbsp;</li>
<li>total positivity,</li>
<li>generalized associahedra,</li>
<li>Poisson geometry, and</li>
<li>Teichm&uuml;ller theory.</li>
</ul>
<p>In the <strong>past 12 months</strong>, there appeared</p>
<ul>
<li>about <strong>35</strong> papers on the <strong>arXiv</strong> having "cluster algebra" in the title,</li>
<li>and about <strong>100</strong> papers having it in the abstract.</li>
</ul>
<h2><strong>2. The cluster complex</strong></h2>
<p>Let $\mathcal{A}$ be a cluster algebra of finite type. The cluster complex $C(\mathcal{A}_W)$ is the simplicial complex with</p>
<ul>
<li>vertices being <strong>cluster variables</strong>,</li>
<li>facets being <strong>clusters</strong>.</li>
</ul>
<p>The <strong>main aim of this talk</strong> is a presentation of</p>
<ul>
<li>a new <strong>combinatorial description of the cluster complex</strong> in finite types<br />(joint work with <strong>C. Ceballos</strong> and <strong>J.-P. Labb&eacute;</strong>)<br /><br /></li>
<li>a construction of a polytope which <strong>graph is exactly the exchange graph</strong> of the cluster algebra / cluster complex<br />(joint work with <strong>V. Pilaud</strong>)</li>
</ul>
<p>Both constructions together provide a <strong>completely new approach to cluster complexes in finite types</strong>.</p>
<h2>4. Subword complexes</h2>
<p><span style="font-size: small;">(introduced by A. Knutson and E. Miller 2003)</span></p>
<p><span style="font-size: small;">Let</span></p>
<ul>
<li><span style="font-size: small;">$(W,S)$ be a <strong>Coxeter system</strong>,</span></li>
<li><span style="font-size: small;">$Q$ be a (not necessarily reduced)&nbsp; <strong>word</strong> in $S$, and</span></li>
<li><span style="font-size: small;">$w$ be an <strong>element</strong> in $W$.</span></li>
</ul>
<p><span style="font-size: small;">The subword complex $\Delta(Q,w)$ is the simplicial complex with</span></p>
<ul>
<li><span style="font-size: small;">vertices being (positions of) <strong>letters</strong> in $Q$,</span></li>
<li><span style="font-size: small;">facets being <strong>complements of </strong>subwords of $Q$ that are<strong> reduced expressions</strong> for $w$.<br /></span></li>
</ul>
<div id="_mcePaste" style="position: absolute; left: -10000px; top: 5px; width: 1px; height: 1px; overflow: hidden;"><span style="font-size: small;">(introduced by S. Fomin and A. Zelevinsky 2001)</span></div>

{{{id=11|
W = CoxeterGroup(['A',2],index_set=[1,2])
Q = [1,2,1,2,1]
w = W.w0
S = SubwordComplex(Q,w)
S.facets()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\left(1, 2\right), \left(0, 4\right), \left(2, 3\right), \left(0, 1\right), \left(3, 4\right)\right\}</span></html>
}}}

<p><span style="font-size: large;"><strong><span style="font-size: x-large;">Theorem</span> (Ceballos-Labb&eacute;-S.)</strong></span></p>
<p>The cluster complex can be obtained as a subword complex for a well-chosen word $Q = {\bf cw}_\circ$ and the longest element $w_\circ \in W$,</p>
<p>$$C(\mathcal{A}_W) \cong \Delta(Q,w_\circ).$$</p>
<h2>5. Generalized associahedra</h2>

{{{id=20|
W = CoxeterGroup(['A',3],index_set=[1,2,3])
Q = [2,3,1,3,2,1,2,3,1]
w = W.w0
S = SubwordComplex(Q,w)
S.facets()
///
<html><span class="math">\newcommand{\Bold}[1]{\mathbf{#1}}\left\{\left(1, 2, 8\right), \left(3, 6, 8\right), \left(3, 4, 5\right), \left(2, 3, 4\right), \left(1, 2, 4\right), \left(1, 4, 5\right), \left(1, 6, 8\right), \left(2, 3, 8\right), \left(1, 5, 6\right), \left(3, 5, 6\right)\right\}</span></html>
}}}

<p><span style="font-size: x-large;">Theorem</span><span style="font-weight: bold; font-size: large;"> (Pilaud-S.)</span></p>
<ul>
<li>The <strong>brick polytope</strong> $\mathcal{B}({\bf cw}_\circ)$ <strong>realizes</strong> the subword complex $S({\bf cw}_\circ)$ and thus<strong> the cluster complex </strong>$C(\mathcal{A}_W)$.</li>
<li>The above realization is an <strong>affine translation</strong> of the polytopal  realizations of <strong>Hohlweg-Lange-Thomas</strong> and of<strong> Chapoton-Fomin-Zelevinsky</strong>.</li>
</ul>
<p><span style="font-size: x-large;"><strong>Corollary</strong></span></p>
<p>The brick polytope $\mathcal{B}({\bf cw}_\circ)$</p>
<ul>
<li>gives an explicit way to <strong>realize generalized associahedra</strong>,</li>
<li>gives a<strong> vertex </strong>and a<strong> facet description</strong> of the generalized associahedra,</li>
<li>gives a <strong>natural Minkowski decomposition</strong> of generalized associahedra into <strong>Coxeter matroid polytopes</strong>.</li>
</ul>

{{{id=24|

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}}}