## Y-systems and generalized associahedra

Venue: | Ann. of Math |

Citations: | 59 - 8 self |

### BibTeX

@ARTICLE{Fomin_y-systemsand,

author = {Sergey Fomin and Nathan Reading},

title = {Y-systems and generalized associahedra},

journal = {Ann. of Math},

year = {},

volume = {158},

pages = {977--1018}

}

### OpenURL

### Abstract

Root systems and generalized associahedra 1 Root systems and generalized associahedra 3

### Citations

459 |
Reflection groups and Coxeter groups
- Humphreys
(Show Context)
Citation Context ...courage the reader to try proving the lemmas, or at least get an idea of why they are true. For additional information on root systems, reflection groups and Coxeter groups, the reader is referred to =-=[9, 25, 34]-=-. For basic definitions related to convex polytopes and lattice theory, see [57] and [31], respectively. Primary sources on generalized associahedra and cluster combinatorics are [13, 19, 21]. Introdu... |

373 |
Representation theory, A first course
- Fulton, Harris
- 1991
(Show Context)
Citation Context ...courage the reader to try proving the lemmas, or at least get an idea of why they are true. For additional information on root systems, reflection groups and Coxeter groups, the reader is referred to =-=[9, 25, 34]-=-. For basic definitions related to convex polytopes and lattice theory, see [57] and [31], respectively. Primary sources on generalized associahedra and cluster combinatorics are [13, 19, 21]. Introdu... |

336 | Infinite Dimensional Lie Algebras, 3rd ed - Kac - 1990 |

319 |
Polylogarithms and Associated Functions
- Lewin
- 1981
(Show Context)
Citation Context ...elated to (and easily deduced from) the famous “pentagonal identity” for the dilogarithm function, first obtained by W. Spence (1809) and rediscovered by Abel (1830) and C. H. Hill (1830). See, e.g., =-=[37]-=-.s6 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA reflections generate finite dihedral groups? To keep things simple, we set s = s1 and confine the choice of s ′ to maps of the form s ′ � � � � x x : ↦−→ ... |

319 |
Homotopy associativity of H-spaces
- Stasheff
- 1963
(Show Context)
Citation Context ...l property: this exchange graph is the 1-skeleton of a convex polytope, the 3-dimensional associahedron. (Sometimes it is also called the Stasheff polytope, after J. Stasheff, who first defined it in =-=[52]-=-.) Figure 3.5 shows a polytopal realization of this associahedron.sLECTURE 3. ASSOCIAHEDRA AND MUTATIONS 27 Figure 3.3. The exchange graph for triangulations of a pentagon. Figure 3.4. The exchange gr... |

224 |
Regular polytopes
- Coxeter
- 1973
(Show Context)
Citation Context ...re no other regular polytopes besides the ones described in Section 1.2. In particular, there are no “exceptional” regular polytopes beyond dimension 4: only simplices, cubes, and crosspolytopes. See =-=[14]-=-. Lie algebras The original motivation for the Cartan-Killing classification of root systems came from Lie theory. Complex finite-dimensional simple Lie algebras correspond naturally, and one-to-one, ... |

113 |
On the self-linking of knots
- Bott, Taubes
- 1994
(Show Context)
Citation Context ... are the Narayana numbers (see Section 5.2). This allows one to calculate the number of faces of each dimension. 3.2. Cyclohedron The n-dimensional cyclohedron (also known as the Bott-Taubes polytope =-=[8]-=-) is constructed similarly to the associahedron using centrally-symmetric triangulations of a regular (2n + 2)-gon. Each edge of the cyclohedron represents either a diagonal flip involving two diamete... |

112 | Non-crossing partitions for classical reflection groups, Discrete Math
- Reiner
- 1997
(Show Context)
Citation Context ...set of Φ is the partial order on the set of positive roots Φ+ such that β ≤ γ if and only if γ − β is a nonnegative (integer) linear combination of simple roots. See Figures 5.2 and 5.3. Theorem 5.2 (=-=[11, 43, 46]-=-). The number of antichains (i.e., sets of pairwise non-comparable elements) in the root poset of Φ is equal to N(Φ). α1 α1 + α2 α2 α1 α1 + α2 2α1 + α2 α2 α1 α1 + α2 2α1 + α2 Figure 5.2. The root pose... |

105 |
Lie groups and Lie algebras, Chapters 4-6
- Bourbaki
- 2002
(Show Context)
Citation Context ...courage the reader to try proving the lemmas, or at least get an idea of why they are true. For additional information on root systems, reflection groups and Coxeter groups, the reader is referred to =-=[9, 25, 34]-=-. For basic definitions related to convex polytopes and lattice theory, see [57] and [31], respectively. Primary sources on generalized associahedra and cluster combinatorics are [13, 19, 21]. Introdu... |

103 | Cluster Algebras II: Finite type classification
- Fomin, Zelevinsky
(Show Context)
Citation Context ...ferred to [9, 25, 34]. For basic definitions related to convex polytopes and lattice theory, see [57] and [31], respectively. Primary sources on generalized associahedra and cluster combinatorics are =-=[13, 19, 21]-=-. Introductory surveys on cluster algebras were given in [22, 55, 56]. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA. E-mail address: fomin@umich.edu, nreading@umich... |

99 |
Cluster algebras I
- Fomin, Zelevinsky
(Show Context)
Citation Context ...the simplicial complex dual to the corresponding generalized associahedron. Example 5.8. The f-vector of the simplicial complex dual to the 3-dimensional cyclohedron (the associahedron of type B3) is =-=(1, 12, 30, 20)-=-. The corresponding h-vector is (1, 9, 9, 1). In general, the Narayana numbers of type Bn are the squares of entries of Pascal’s triangle: Nk(Bn) = � � n 2. k It is easy to see that the entries of an ... |

97 | Conjectures on the quotient ring by diagonal invariants
- Haiman
- 1994
(Show Context)
Citation Context ...AHEDRA W-orbits in a discrete torus The reflection group W acts on the root lattice Q = ZΦ, hence on the “discrete torus” Q/(h + 1)Q obtained as a quotient of Q by its subgroup (h + 1)Q. Theorem 5.4 (=-=[32]-=-). The number of W-orbits in Q/(h + 1)Q is equal to N(Φ). Figures 5.5 and 5.6 illustrate these orbits in types A2 and B2, where h = 3 and h = 4, respectively. Each figure shows the reflection lines of... |

82 |
On the number of reduced decompositions of elements of Coxeter groups
- Stanley
- 1984
(Show Context)
Citation Context ...path along edges from 1 to w which moves up in a monotone fashion. There are 16 such paths from 1 to w◦ in the A3-permutohedron; cf. Example 2.12. The following beautiful formula is due to R. Stanley =-=[49]-=-. Theorem 2.15. The number of reduced words for w◦ in the reflection group An is � � n+1 2 ! 1n3n−15n−2 . · · · (2n − 1) 1 1 An Archimedean solid is a non-regular polytope whose all facets are regular... |

74 |
Enumerative Combinatorics Vol 2
- Stanley
- 1999
(Show Context)
Citation Context ...bed above are enumerated 1 2n+2 by the Catalan numbers n+2 n+1 . There are a great many families of combinatorial objects enumerated by the Catalan numbers; more than a hundred of those are listed in =-=[50]-=-. This list includes: ballot sequences; Young diagrams and tableaux satisfying certain restrictions; noncrossing partitions; trees of various kinds; Dyck paths; permutations avoiding patterns of lengt... |

70 | Goncharov, “Moduli Spaces of Local Systems and Higher Teichmüller Theory
- Fock, B
(Show Context)
Citation Context ...y passing from Euclidean to hyperbolic geometry, where an analogue of Ptolemy’s Theorem holds, and where one can “cook up” the required additional degrees of freedom. For much more on this topic, see =-=[24, 29]-=-.s38 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA Plücker coordinates on the Grassmannian Gr(2, n + 3) Take a 2 × (n+3) matrix z = (zij). For any k, l ∈ [n + 3], k < l, let us denote by � � z1k z1l Pkl =... |

68 |
The associahedron and triangulations of the n-gon
- Lee
- 1989
(Show Context)
Citation Context ...n be realized as a boundary of an n-dimensional convex polytope. Theorem 3.4 (or its equivalent reformulations) were proved independently by J. Milnor, M. Haiman, and C. W. Lee (first published proof =-=[36]-=-). This theorem also follows as a special case of the very general theory of secondary polytopes developed by I. M. Gelfand, M. Kapranov and A. Zelevinsky [30]. Definition 3.5 (The associahedron). The... |

67 | Cluster Algebras III: Upper bounds and double Bruhat cells, preprint ArXiv:math.RT/0305434
- Berenstein, Fomin, et al.
(Show Context)
Citation Context ..., among other fields. All these motivations and applications will remain behind the scenes in these lectures. Most of this lecture is based on [19, 20, 21]. Sections 4.4 and 4.5 are based on [13] and =-=[4, 23]-=-, respectively. 4.1. Seeds and clusters Consider a diagonal flip that transforms a triangulation T of a convex (n + 3)-gon into another triangulation T ′ , as shown in Figure 3.14. The corresponding e... |

61 |
A remarkable q, t-Catalan sequence and q-Lagrange inversion
- Garsia, Haiman
- 1996
(Show Context)
Citation Context ...with their standard q-analogues. A difn+2 n+1 n+1 ferent answer is obtained while counting order ideals in the root poset of type An by the cardinality of an ideal. For more on these q-analogues, see =-=[26, 27, 51]-=-. We will focus on a third�q-analogue �� � that is related to the Narayana numbers, 1 n+1 n+1 defined by the formula n+1 k k+1 . The Narayana numbers form a triangle shown on the right in Figure 5.10.... |

48 |
q-Catalan numbers
- Fürlinger, Hofbauer
- 1985
(Show Context)
Citation Context ...with their standard q-analogues. A difn+2 n+1 n+1 ferent answer is obtained while counting order ideals in the root poset of type An by the cardinality of an ideal. For more on these q-analogues, see =-=[26, 27, 51]-=-. We will focus on a third�q-analogue �� � that is related to the Narayana numbers, 1 n+1 n+1 defined by the formula n+1 k k+1 . The Narayana numbers form a triangle shown on the right in Figure 5.10.... |

47 |
The dual braid monoid, Ann
- Bessis
- 2003
(Show Context)
Citation Context ...n arbitrary order) of the generators in S. Thus, c is a Coxeter element in W, in the broader sense of the notion alluded to in a footnote in Section 2.5. The non-crossing partition lattice for W (see =-=[7, 10]-=-) is the interval [1, c] in the partial order (W, �) defined above. It is a classical result that all Coxeter elements are conjugate to each other. Since the set of all reflections is fixed under conj... |

45 | Polytopal realizations of generalized associahedra
- Chapoton, Fomin, et al.
(Show Context)
Citation Context ...ferred to [9, 25, 34]. For basic definitions related to convex polytopes and lattice theory, see [57] and [31], respectively. Primary sources on generalized associahedra and cluster combinatorics are =-=[13, 19, 21]-=-. Introductory surveys on cluster algebras were given in [22, 55, 56]. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA. E-mail address: fomin@umich.edu, nreading@umich... |

44 |
Ad-nilpotent ideals of a Borel subalgebra
- Cellini, Papi
(Show Context)
Citation Context ...set of Φ is the partial order on the set of positive roots Φ+ such that β ≤ γ if and only if γ − β is a nonnegative (integer) linear combination of simple roots. See Figures 5.2 and 5.3. Theorem 5.2 (=-=[11, 43, 46]-=-). The number of antichains (i.e., sets of pairwise non-comparable elements) in the root poset of Φ is equal to N(Φ). α1 α1 + α2 α2 α1 α1 + α2 2α1 + α2 α2 α1 α1 + α2 2α1 + α2 Figure 5.2. The root pose... |

42 | Total Positivity: tests and parametrizations
- Fomin, Zelevinsky
(Show Context)
Citation Context ... knowledge of Coxeter groups and root systems. Some of the figures in these notes are inspired by figures produced by Satyan Devadoss, Vic Reiner and Rodica Simion. Several figures were borrowed from =-=[13, 19, 20, 21, 23]-=-.s1.1. The pentagon recurrence LECTURE 1. REFLECTIONS AND ROOTS 5 LECTURE 1 Reflections and roots Consider a sequence f1, f2, f3, . . . defined recursively by f1 = x, f2 = y, and (1) fn+1 = fn + 1 Thu... |

38 | Noncrossing partitions for the group Dn
- Athanasiadis, Reiner
(Show Context)
Citation Context ...various combinatorial objects related to the root system Φ. Below in this section, we briefly describe several families of objects counted by N(Φ). We refer the reader to the introductory sections of =-=[1, 3, 2, 12, 39]-=- for the history of research in this area, for further details and references, and for numerous generalizations and connections. The numbers N(Φ) seem to have first appeared in D. Djoković’s work [18]... |

36 | Enumerative properties of generalized associahedra
- Chapoton
(Show Context)
Citation Context ...various combinatorial objects related to the root system Φ. Below in this section, we briefly describe several families of objects counted by N(Φ). We refer the reader to the introductory sections of =-=[1, 3, 2, 12, 39]-=- for the history of research in this area, for further details and references, and for numerous generalizations and connections. The numbers N(Φ) seem to have first appeared in D. Djoković’s work [18]... |

34 | Cluster algebras and Weil-Petersson forms
- Gekhtman, Shapiro, et al.
(Show Context)
Citation Context ...y passing from Euclidean to hyperbolic geometry, where an analogue of Ptolemy’s Theorem holds, and where one can “cook up” the required additional degrees of freedom. For much more on this topic, see =-=[24, 29]-=-.s38 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA Plücker coordinates on the Grassmannian Gr(2, n + 3) Take a 2 × (n+3) matrix z = (zij). For any k, l ∈ [n + 3], k < l, let us denote by � � z1k z1l Pkl =... |

28 | On the Charney-Davis and Neggers-Stanley conjectures
- Reiner, Welker
- 2005
(Show Context)
Citation Context ...ns of N(Φ) given in Section 5.1. These enumerative results are listed in Theorem 5.9 below; we elaborate on the items in the theorem in subsequent comments. Theorem 5.9 is a combination of results in =-=[3, 19, 39, 44, 48]-=-; see [3] for a historical overview, and for further generalizations.sLECTURE 5. ENUMERATIVE PROBLEMS 61 n� � �� � 1 n + 1 n + 1 N(An, q) = q n + 1 k k + 1 k=0 k n� � �2 n N(Bn, q) = q k k=0 k N(Dn, q... |

27 | Algebraic operads
- Loday, Vallette
(Show Context)
Citation Context ...can be realized as a convex polytope—is essentially equivalent to Theorem 3.4.sLECTURE 3. ASSOCIAHEDRA AND MUTATIONS 31 Associahedra play an important role in homotopy theory and the study of operads =-=[53]-=-, in the analysis of real moduli/configuration spaces [16], and other branches of mathematics. In these notes, we restrict our attention to the combinatorial aspects of the associahedra. An n-dimensio... |

25 |
Ad-nilpotent ideals of a Borel subalgebra: generators and duality
- Panyushev
(Show Context)
Citation Context ...various combinatorial objects related to the root system Φ. Below in this section, we briefly describe several families of objects counted by N(Φ). We refer the reader to the introductory sections of =-=[1, 3, 2, 12, 39]-=- for the history of research in this area, for further details and references, and for numerous generalizations and connections. The numbers N(Φ) seem to have first appeared in D. Djoković’s work [18]... |

25 | B-stable ideals in the nilradical of a Borel subalgebra
- Sommers
(Show Context)
Citation Context ...ns of N(Φ) given in Section 5.1. These enumerative results are listed in Theorem 5.9 below; we elaborate on the items in the theorem in subsequent comments. Theorem 5.9 is a combination of results in =-=[2, 19, 39, 44, 48]-=-; see [2] for a historical overview, and for further generalizations.sLECTURE 5. ENUMERATIVE PROBLEMS 61 n� � �� � 1 n + 1 n + 1 N(An, q) = q n + 1 k k + 1 k=0 k n� � �2 n N(Bn, q) = q k k=0 k N(Dn, q... |

24 |
Orderings of Coxeter groups, Combinatorics and Algebra
- Björner
- 1984
(Show Context)
Citation Context ...of u (necessarily by 1). Lemma 2.13 (see also the paragraph that follows it) implies that the Hasse diagram of the weak order can be identified with the 1-skeleton of a W-permutohedron. Theorem 5.12 (=-=[6]-=-). The weak order on a finite Coxeter group is a lattice. Example 5.13. The weak order of type An can be described in the language of permutations of [n+1], written in one-line notation. Permutation v... |

23 | Cambrian lattices
- Reading
(Show Context)
Citation Context ... 63 Figure 5.14. The associahedron of type H3 H3 H4 I2(m) 32 280 m + 2 Figure 5.15. The numbers N(Φ) in non-crystallographic cases 5.4. Lattice congruences and the weak order This section is based on =-=[41]-=-. Its main goal is to establish a relationship between two fans associated with a root system Φ and the corresponding reflection group W: • the Coxeter fan created by (the regions of) the Coxeter arra... |

20 |
The number of ⊕-sign types
- Shi
- 1997
(Show Context)
Citation Context ...set of Φ is the partial order on the set of positive roots Φ+ such that β ≤ γ if and only if γ − β is a nonnegative (integer) linear combination of simple roots. See Figures 5.2 and 5.3. Theorem 5.2 (=-=[11, 43, 46]-=-). The number of antichains (i.e., sets of pairwise non-comparable elements) in the root poset of Φ is equal to N(Φ). α1 α1 + α2 α2 α1 α1 + α2 2α1 + α2 α2 α1 α1 + α2 2α1 + α2 Figure 5.2. The root pose... |

17 | Cluster Algebras: Notes for the CDM-03 conference - Fomin, Zelevinsky - 2003 |

15 | Explicit presentations for the dual braid monoids
- Picantin
- 2002
(Show Context)
Citation Context ...lections is fixed under conjugation, it follows that different choices of c yield isomorphic posets. (These posets are lattices, which is a non-trivial theorem.) The following theorem was obtained in =-=[7, 40]-=-. A version for the classical types ABCD appeared earlier in [43]. Theorem 5.5. Let W be the reflection group associated with a finite root system Φ. Then the non-crossing partition lattice for W has ... |

15 |
A type-B associahedron, Adv
- Simion
(Show Context)
Citation Context ...sional cyclohedra respectively. As these figures suggest, the cyclohedron is a convex polytope for any n. Explicit polytopal realizations of cyclohedra were constructed by M. Markl [38] and R. Simion =-=[47]-=-. Each face of a cyclohedron is a product of smaller cyclohedra and associahedra. Figure 3.8. The 2-dimensional cyclohedron Further details about the combinatorics of cyclohedra, and about their appea... |

14 |
K(π, 1)’s for Artin groups of finite type, Geom. Dedicata 94
- Brady, Watt
- 2002
(Show Context)
Citation Context ...n arbitrary order) of the generators in S. Thus, c is a Coxeter element in W, in the broader sense of the notion alluded to in a footnote in Section 2.5. The non-crossing partition lattice for W (see =-=[7, 10]-=-) is the interval [1, c] in the partial order (W, �) defined above. It is a classical result that all Coxeter elements are conjugate to each other. Since the set of all reflections is fixed under conj... |

14 |
From Littlewood-Richardson coefficients to cluster algebras in three lectures, in: Symmetric Functions 2001: Surveys of Developments and Perspectives, S.Fomin (Ed.), 253-273
- Zelevinsky
- 2002
(Show Context)
Citation Context ...pes and lattice theory, see [57] and [31], respectively. Primary sources on generalized associahedra and cluster combinatorics are [13, 19, 21]. Introductory surveys on cluster algebras were given in =-=[22, 55, 56]-=-. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA. E-mail address: fomin@umich.edu, nreading@umich.edu. This work was partially supported by NSF grants DMS-0245385 (S.... |

13 |
Athanasiadis, Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes
- A
(Show Context)
Citation Context |

11 | Equivariant fiber polytopes
- Reiner
(Show Context)
Citation Context ... 5.15 has been proved in types An and Bn. The proof makes explicit the combinatorics of the Cambrian congruence and connects it to constructions given by Billera and Sturmfels [5] (type A) and Reiner =-=[42]-=- (type B). The conjecture implies in particular that the Hasse diagram of the quotient of the weak order by the Cambrian congruence (called the Cambrian lattice) is isomorphic to the 1-skeleton of the... |

9 | Combinatorial equivalence of real moduli spaces
- Devadoss
(Show Context)
Citation Context ...lent to Theorem 3.4.sLECTURE 3. ASSOCIAHEDRA AND MUTATIONS 31 Associahedra play an important role in homotopy theory and the study of operads [53], in the analysis of real moduli/configuration spaces =-=[16]-=-, and other branches of mathematics. In these notes, we restrict our attention to the combinatorial aspects of the associahedra. An n-dimensional polytope is called simple if every vertex is incident ... |

8 |
Iterated fiber polytopes, Mathematika 41
- Billera, Sturmfels
- 1994
(Show Context)
Citation Context ...n congruence. Conjecture 5.15 has been proved in types An and Bn. The proof makes explicit the combinatorics of the Cambrian congruence and connects it to constructions given by Billera and Sturmfels =-=[5]-=- (type A) and Reiner [42] (type B). The conjecture implies in particular that the Hasse diagram of the quotient of the weak order by the Cambrian congruence (called the Cambrian lattice) is isomorphic... |

8 |
On conjugacy classes of elements of finite order in compact or complex semisimple Lie groups
- Djoković
- 1980
(Show Context)
Citation Context ... 39] for the history of research in this area, for further details and references, and for numerous generalizations and connections. The numbers N(Φ) seem to have first appeared in D. Djoković’s work =-=[18]-=- on enumeration of conjugacy classes of elements of finite order in Lie groups. 53s54 FOMIN AND READING, ROOTS AND ASSOCIAHEDRA Antichains in the root poset (non-nesting partitions) The root poset of ... |

7 |
Athanasiadis, On a refinement of the generalized Catalan numbers for Weyl groups
- A
(Show Context)
Citation Context |

7 |
General lattice theory, 2nd edition
- Grätzer
(Show Context)
Citation Context ...additional information on root systems, reflection groups and Coxeter groups, the reader is referred to [9, 25, 34]. For basic definitions related to convex polytopes and lattice theory, see [57] and =-=[31]-=-, respectively. Primary sources on generalized associahedra and cluster combinatorics are [13, 19, 21]. Introductory surveys on cluster algebras were given in [22, 55, 56]. Department of Mathematics, ... |

6 |
The ubiquity of Coxeter-Dynkin diagrams (an introduction to the A-D-E problem). Nieuw
- Hazewinkel, Hesselink, et al.
- 1977
(Show Context)
Citation Context ...nces therein. Et cetera And the list goes on: simple singularities, finite subgroups of SU(2), symmetric matrices with nonnegative integer entries and eigenvalues between −2 and 2, etc. For more, see =-=[28, 33, 58]-=-. In Section 4.2, we will present yet another classification that is parallel to Cartan-Killing: the classification of the cluster algebras of finite type. 2.4. Reduced words and permutohedra Each ele... |

6 |
On the Charney-Davis and Neggers-Stanley
- Reiner, Welker
(Show Context)
Citation Context ...ns of N(Φ) given in Section 5.1. These enumerative results are listed in Theorem 5.9 below; we elaborate on the items in the theorem in subsequent comments. Theorem 5.9 is a combination of results in =-=[2, 19, 39, 44, 48]-=-; see [2] for a historical overview, and for further generalizations.sLECTURE 5. ENUMERATIVE PROBLEMS 61 n� � �� � 1 n + 1 n + 1 N(An, q) = q n + 1 k k + 1 k=0 k n� � �2 n N(Bn, q) = q k k=0 k N(Dn, q... |

6 |
Dynkin diagrams and the representation theory of algebras
- Reiten
- 1997
(Show Context)
Citation Context ... is indecomposable if it cannot be obtained as a nontrivial direct sum. By Gabriel’s Theorem, a quiver is of finite type if and only if its underlying graph is a Dynkin diagram of type A, D or E. See =-=[45]-=- and references therein. Et cetera And the list goes on: simple singularities, finite subgroups of SU(2), symmetric matrices with nonnegative integer entries and eigenvalues between −2 and 2, etc. For... |

6 |
Cluster algebras: notes for 2004
- Zelevinsky
(Show Context)
Citation Context ...pes and lattice theory, see [57] and [31], respectively. Primary sources on generalized associahedra and cluster combinatorics are [13, 19, 21]. Introductory surveys on cluster algebras were given in =-=[22, 55, 56]-=-. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA. E-mail address: fomin@umich.edu, nreading@umich.edu. This work was partially supported by NSF grants DMS-0245385 (S.... |

5 | Reflection groups. A contribution to the Handbook of Algebra. arXiv:math/0311012
- Geck, Malle
(Show Context)
Citation Context ...nces therein. Et cetera And the list goes on: simple singularities, finite subgroups of SU(2), symmetric matrices with nonnegative integer entries and eigenvalues between −2 and 2, etc. For more, see =-=[28, 33, 58]-=-. In Section 4.2, we will present yet another classification that is parallel to Cartan-Killing: the classification of the cluster algebras of finite type. 2.4. Reduced words and permutohedra Each ele... |