On the enumeration of positive cells in generalized cluster complexes and Catalan hyperplane arrangements
Christos A. Athanasiadis1
and Eleni Tzanaki2
1University of Athens Department of Mathematics (Division of Algebra-Geometry) Panepistimioupolis Athens 15784 Greece
2University of Crete Department of Mathematics 71409 Heraklion Crete Greece
2University of Crete Department of Mathematics 71409 Heraklion Crete Greece
DOI: 10.1007/s10801-006-8348-8
Abstract
Let Φ be an irreducible crystallographic root system with Weyl group W and coroot lattice \check Q \check{Q} , spanning a Euclidean space V. Let m be a positive integer and A m F {\mathcal A}^{m}_{Φ} be the arrangement of hyperplanes in V of the form ( a, x) = k (α, x) = k for a Ĩ F α\in Φ and k = 0, 1,..., m k = 0, 1,\dots,m . It is known that the number N + ( F, m) N^+ (Φ, m) of bounded dominant regions of A m F {\mathcal A}^{m}_{Φ} is equal to the number of facets of the positive part D m + ( F) Δ^m_+ (Φ) of the generalized cluster complex associated to the pair ( F, m) (Φ, m) by S. Fomin and N. Reading.
We define a statistic on the set of bounded dominant regions of A m F {\mathcal A}^{m}_{Φ} and conjecture that the corresponding refinement of N + ( F, m) N^+ (Φ, m) coincides with the h h-vector of D m + ( F) Δ^m_+ (Φ) . We compute these refined numbers for the classical root systems as well as for all root systems when m = 1 and verify the conjecture when Φ has type A, B or C and when m = 1. We give several combinatorial interpretations to these numbers in terms of chains of order ideals in the root poset of Φ , orbits of the action of W on the quotient \check Q / ( mh -1) \check Q \check{Q} / \, (mh-1) \, \check{Q} and coroot lattice points inside a certain simplex, analogous to the ones given by the first author in the case of the set of all dominant regions of A m F {\mathcal A}^{m}_{Φ} . We also provide a dual interpretation in terms of order filters in the root poset of Φ in the special case m = 1.
Pages: 355–375
Keywords: keywords Catalan arrangement; bounded region; generalized cluster complex; positive part; $h$-vector
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References
1. C.A. Athanasiadis, Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes, Bull. London Math. Soc. 36 (2004), 294-302.
2. C.A. Athanasiadis, On a refinement of the generalized Catalan numbers for Weyl groups, Trans. Amer. Math. Soc. 357 (2005), 179-196. Springer
3. C.A. Athanasiadis and V. Reiner, Noncrossing partitions for the group Dn, SIAM J. Discrete Math. 18 (2004), 397-417.
4. N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6, Springer-Verlag, Berlin, 2002.
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9. S. Fomin and A.V. Zelevinsky, Y -systems and generalized associahedra, Ann. of Math. 158 (2003), 977-1018.
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11. J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, Cambridge, England, 1990.
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13. D.I. Panyushev, Ad-nilpotent ideals of a Borel subalgebra: generators and duality, J. Algebra 274 (2004), 822-846.
14. D.I. Panyushev, Two covering polynomials of a finite poset, with applications to root systems and adnilpotent ideals, preprint, 2005, 20pp (ArXiV preprint math.CO/0502386).
15. J.-Y. Shi, Alcoves corresponding to an affine Weyl group, J. London Math. Soc. 35 (1987), 42-55.
16. E. Sommers, B-stable ideals in the nilradical of a Borel subalgebra, Canad. Math. Bull. 48 (2005), 460-472.
17. R.P. Stanley, Enumerative Combinatorics, vol. 1, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1986; second printing, Cambridge University Press, Cambridge, 1997.
18. E. Tzanaki, Polygon dissections and some generalizations of cluster complexes, J. Combin. Theory Series A (to appear).
2. C.A. Athanasiadis, On a refinement of the generalized Catalan numbers for Weyl groups, Trans. Amer. Math. Soc. 357 (2005), 179-196. Springer
3. C.A. Athanasiadis and V. Reiner, Noncrossing partitions for the group Dn, SIAM J. Discrete Math. 18 (2004), 397-417.
4. N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4-6, Springer-Verlag, Berlin, 2002.
5. F. Chapoton, Enumerative properties of generalized associahedra, Sémin. Loth. de Combinatoire 51 (2004), Article B51b, 16pp (electronic).
6. S. Fomin and N. Reading, Root systems and generalized associahedra, in Geometric Combinatorics, IAS/Park City Mathematics Series (to appear).
7. S. Fomin and N. Reading, Generalized cluster complexes and Coxeter combinatorics, incomplete preliminary draft of January 6, 2005, 30pp.
8. S. Fomin and N. Reading, Generalized cluster complexes and Coxeter combinatorics, Int. Math. Res. Not. 44 (2005), 2709-2757.
9. S. Fomin and A.V. Zelevinsky, Y -systems and generalized associahedra, Ann. of Math. 158 (2003), 977-1018.
10. M.D. Haiman, Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), 17-76.
11. J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, Cambridge, England, 1990.
12. R. Marsh, M. Reineke and A. Zelevinsky, Generalized associahedra via quiver representations, Trans. Amer. Math. Soc. 355 (2003), 4171-4186.
13. D.I. Panyushev, Ad-nilpotent ideals of a Borel subalgebra: generators and duality, J. Algebra 274 (2004), 822-846.
14. D.I. Panyushev, Two covering polynomials of a finite poset, with applications to root systems and adnilpotent ideals, preprint, 2005, 20pp (ArXiV preprint math.CO/0502386).
15. J.-Y. Shi, Alcoves corresponding to an affine Weyl group, J. London Math. Soc. 35 (1987), 42-55.
16. E. Sommers, B-stable ideals in the nilradical of a Borel subalgebra, Canad. Math. Bull. 48 (2005), 460-472.
17. R.P. Stanley, Enumerative Combinatorics, vol. 1, Wadsworth & Brooks/Cole, Pacific Grove, CA, 1986; second printing, Cambridge University Press, Cambridge, 1997.
18. E. Tzanaki, Polygon dissections and some generalizations of cluster complexes, J. Combin. Theory Series A (to appear).
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