Cluster algebra: Difference between revisions

Suppose that ''F'' is an [[integral domain]], such as the [[Field (mathematics)|field]] '''Q'''(''x''₁,...,''x''_''n'') of [[rational functionsfunction]]s in ''n'' variables over the [[rational number|rational numbers]]s '''Q'''.

A '''cluster''' of '''rank''' ''n'' consists of a set of ''n'' elements {''x'', ''y'', ...} of ''F'', usually assumed to be an [[Algebraic independence|algebraically independent]] set of generators of a [[field extension]] ''F''.

A '''seed''' consists of a cluster {''x'', ''y'', ...} of ''F'', together with an '''exchange matrix''' ''B'' with integer entries ''b''_''x'',''y'' indexed by pairs of elements ''x'', ''y'' of the cluster. The matrix is sometimes assumed to be skew[[Skew-symmetric matrix|skew-symmetric]], so that ''b''_''x'',''y'' = –''b''_''y'',''x'' for all ''x'' and ''y''. More generally the matrix might be skew -symmetrizable, meaning there are positive integers ''d''_''x'' associated with the elements of the cluster such that ''d''_''x''''b''_''x'',''y'' = –''d''_''y''''b''_''y'',''x'' for all ''x'' and ''y''. It is common to picture a seed as a [[quiver (mathematics)|quiver]] withwhose vertices are the generating set, by drawing ''b''_''x'',''y'' arrows from ''x'' to ''y'' if this number is positive. When ''b''_''x'',''y'' is skew symmetrizable the quiver has no loops or 2-cycles.

A '''mutation''' of a seed, depending on a choice of vertex ''y'' of the cluster, is a new seed given by a generalization of [[Tilting theory|tilting]] as follows. Exchange the values of ''b''_''x'',''y'' and ''b''_''y'',''x'' for all ''x'' in the cluster. If ''b''_''x'',''y'' > 0 and ''b''_''y'',''z'' > 0 then replace ''b''_''x'',''z'' by ''b''_''x'',''y''''b''_''y'',''z'' + ''b''_''x'',''z''. If ''b''_''x'',''y'' < 0 and ''b''_''y'',''z'' < 0 then replace ''b''_''x'',''z'' by -''b''_''x'',''y''''b''_''y'',''z'' + ''b''_''x'',''z''. If ''b''_''x'',''y'' ''b''_''y'',''z'' <=≤ 0 then do not change ''b''_''x'',''z''. Finally replace ''y'' by a new generator ''w'', where

:<math>wy=\prod_{t: \, b_{t,y}>0}t^{b_{t,y}} + \prod_{t: \, b_{t,y}<0}t^{-b_{t,y}}</math>

where the products run through the elements ''t'' in the cluster of the seed such that ''b''_''t'',''y'' is positive or negative respectively. The inverse of a mutation is also a mutation:, in other words,i.e. if ''A'' is a mutation of ''B'', then ''B'' is a mutation of ''A''.

If {''x''} is the cluster of a seed of rank 1, then the only mutation takes this to {2''x''⁻¹}. So a cluster algebra of rank 1 is just a ring ''k''[''x'',''x''⁻¹] of [[Laurent polynomialspolynomial]]s, and it has just two clusters, {''x''} and {2''x''⁻¹}. In particular it is of finite type and is associated with the Dynkin diagram A₁.

Suppose that we start with the cluster {''x''₁, ''x''₂} and take the exchange matrix with ''b''₁₂ = –b₂₁ = 1. Then mutation gives a sequence of variables ''x''₁, ''x''₂, ''x''₃, ''x''₄,... such that the clusters are given by adjacent pairs {''x''_''n'', ''x''_''n''+1}. The variables are related by

:<math>\displaystyle x_{n-1}x_{n+1}=1+x_n,</math>

:<math> x_1, \ x_2, \ x_3 = \frac{1+x_2}{x_1}, \ x_4 = \frac{1+x_3}{x_2}=\frac{1+x_1+x_2}{x_1x_2}, </math>

: <math> x_5 = \frac{1+x_4}{x_3} = \frac{1+x_1}{x_2}, \ x_6 = \frac{1+x_5}{x_4} = x_1, \ x_7 = \frac{1+x_6}{x_5} = x_2, \ \ldots</math>

Suppose we start with the quiver ''x''₁ → ''x''₂ → ''x''₃. Then the 14 clusters are:

For the Grassmannian of planes in ℂ<supmath>''\mathbb{C}^n''</supmath>, the situation is even more simple. In that case, the Plücker coordinates provide all the distinguished elements and the clusters can be completely described using [[Polygon triangulation|triangulation]]s of a [[regular polygon]] with ''n'' vertices. More precisely, clusters are in one-to-one correspondence with triangulations and the distinguished elements are in one-to-one correspondence with diagonals (line segments joining two vertices of the polygon). One can distinguish between diagonals in the boundary, which belong to every cluster, and diagonals in the interior. This corresponds to a general distinction between coefficient variables and cluster variables.

Suppose '''S''' is a [[Compact space|compact]] [[Connected space|connected]] [[Orientability|oriented]] [[Riemann surface]] and '''M''' is a [[Empty set|non-empty]] finite set of points in '''S''' that contains at least one point from each [[Boundary (topology)|boundary]] component of '''S''' (the boundary of '''S''' is not assumed to be either empty or non-empty). The pair ('''S''', '''M''') is often referred to as a ''bordered surface with marked points''. It has been shown by Fomin-Shapiro-Thurston that if '''S''' is not a closed surface, or if '''M''' has more than one point, then the (tagged) arcs on ('''S''', '''M''') parameterize the set of cluster variables of certain cluster algebra ''A''('''S''', '''M'''), which depends only on ('''S''', '''M''') and the choice of some coefficient system, in such a way that the set of (tagged) triangulations of ('''S''', '''M''') is in one-to-one correspondence with the set of clusters of ''A''('''S''', '''M'''), two (tagged) triangulations being related by a ''flip'' if and only if the clusters they correspond to are related by cluster mutation.

For '''<math>G'''</math> a [[reductive group]] such as <math>GL_n</math> with [[Borel subgroup]]s <math>B_\pm</math> then on <math>G^{u,v} = B u B \bigcapcap B_- v B_-</math> (where <math>u,</math> and <math>v</math> are in the [[Weyl group]]) there are cluster coordinate charts depending on reduced word decompositions of <math>u,</math> and <math>v</math>. These are called factorization parameters and their structure is encoded in a wiring diagram. With only <math>B</math> or only <math>B_-</math>, this is [[Bruhat decomposition]].

*{{Citation | last1=Berenstein | first1=Arkady | last2=Fomin | first2=Sergey | last3=Zelevinsky | first3=Andrei | title=Cluster algebras. III. Upper bounds and double Bruhat cells | doi=10.1215/S0012-7094-04-12611-9 | mr=2110627 | year=2005 | journal=[[Duke Mathematical Journal]] | volume=126 | issue=1 | pages=1–52| arxiv=math/0305434 | s2cid=7733033 }}

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*{{Citation | last1=Fomin | first1=Sergey | last2=Zelevinsky | first2=Andrei | title=Cluster algebras. II. Finite type classification | doi=10.1007/s00222-003-0302-y | mr=2004457 | year=2003 | journal=[[Inventiones Mathematicae]] | volume=154 | issue=1 | pages=63–121| arxiv=math/0208229 | bibcode=2003InMat.154...63F | s2cid=14540263 }}

*{{Citation | last1=Fomin | first1=Sergey | last2=Zelevinsky | first2=Andrei | title=Cluster algebras. IV. Coefficients | doi=10.1112/S0010437X06002521 | mr=2295199 | year=2007 | journal=Compositio Mathematica | volume=143 | issue=1 | pages=112–164| arxiv=math/0602259 | s2cid=15744006 }}

*{{citation|MRmr=3155783

|last=Marsh|first= RobertBethany JR.

* {{citation | first = Andrei | last = Zelevinsky | title = What Is . . . a Cluster Algebra? | journal = AMS Notices | volume = 54 | issue = 11 | pages = 1494–1495 | year = 2007 | url = httphttps://www.ams.org/notices/200711/tx071101494p.pdf }}.