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Generalized Catalan numbers and hyperplane arrangements Communicating Mathematics, July, 2007

1. 4. 2. 3. 4. 1. 3. 2. 1. 4. 3. 2. Generalized Catalan numbers and hyperplane arrangements Communicating Mathematics, July, 2007. Cathy Kriloff Idaho State University Supported in part by NSA grant MDA904-03-1-0093 Joint work with Yu Chen, Idaho State University

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Generalized Catalan numbers and hyperplane arrangements Communicating Mathematics, July, 2007

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  1. 1 4 2 3 4 1 3 2 1 4 3 2 Generalized Catalan numbers and hyperplane arrangementsCommunicating Mathematics, July, 2007 Cathy Kriloff Idaho State University Supported in part by NSA grant MDA904-03-1-0093 Joint work with Yu Chen, Idaho State University Journal of Combinatorial Theory – Series A

  2. Outline • Partitions counted by Cat(n) • Real reflection groups • Generalized partitions counted by Cat(W) • Regions in hyperplane arrangements and the dihedral noncrystallographic case

  3. Poset of partitions of [n] • Let P(n)=partitions of [n]={1,2,…,n} • Order by: P1≤P2 if P1 refinesP2 • Same as intersection lattice of Hn={xi=xj | 1≤i<j≤n} in Rn under reverse inclusion • Example: P(3)

  4. Nonnesting partitions of [n] Nonnesting partitions have no nested arcs = NN(n) Examples in P(4): Nonnesting partition of [4] Nesting partition of [4] Noncrossing partitions have no crossing arcs = NC(n) Examples in P(4): Noncrossing partition of [4] Crossing partition of [4]

  5. P(4), NN(4), NC(4) Subposets: • NN(4)=P(4)\ • NC(4)=P(4)\

  6. How many are there? Catalan number See #6.19(pp,uu) in Stanley, Enumerative Combinatorics II, 1999 or www-math.mit.edu/~rstan/ NN(n) Postnikov – 1999 NC(n) Becker - 1948, Kreweras - 1972 These posets are all naturally related to the permutation group Sn

  7. Some crystallographic reflection groups • Symmetries of these shapes are crystallographic reflection groups of types A2, B2, G2 • First two generalize to n-dim simplex and hypercube • Corresponding groups: Sn+1=An and Sn⋉(Z2)n=Bn • (Some crystallographic groups are not symmetries of regular polytopes)

  8. Some noncrystallographic reflection groups • Generalize to 2-dim regular m-gons • Get dihedral groups,I2(m), for any m • Noncrystallographic unless m=3,4,6 (tilings) I2(5) I2(7) I2(8)

  9. Real reflection groups Classification of finite groups generated by reflections = finite Coxeter groups due toCoxeter (1934), Witt (1941) Symmetries of regular polytopes Crystallographic reflection groups =Weyl groups Venn diagram: Drew Armstrong

  10. Root System of type A2 • roots = unit vectors perpendicular to reflecting hyperplanes • simpleroots = basis so each root is positive or negative A2 a1+a2=b2=e1-e3 a2=b3=e2-e3 a1=b1=e1-e2 • ai are simple roots • bi are positive roots • work in plane x1+x2+x3=0 • ei-ej connect to NN(3) since hyperplane xi=xj is (ei-ej)┴

  11. 3 1 2 Root poset in type A2 Root poset for A2 • Express positive j in i basis • Ordering: ≤ if -═cii with ci≥0 • Connect by an edge if comparable • Increases going down • Pick any set of incomparable roots (antichain), , and form its ideal=   for all  • Leave off bs, just write indices Antichains (ideals) for A2 1 (2) 3 1 (2) (2) 3 2 

  12. NN(n) as antichains Let e1,e2,…,en be an orthonormal basis of Rn n=3, type A2 Subposet of intersection lattice of hyperplane arrangement {xi-xj=0 | 1≤i<j≤n} in type An-1, {<x,bi>=0 | 1≤j≤n} in general Antichains (ideals) in Int(n-1) in type An-1 (Stanley-Postnikov 6.19(bbb)), root poset in general

  13. Case when n=4 Using antichains/ideals in the root poset excludes {e1-e4,e2-e3}

  14. Generalized Catalan numbers • For W=crystallographic reflection group define NN(W) to be antichains in the root poset (Postnikov)Get |NN(W)|=Cat(W)= (h+di)/|W|, where h = Coxeter number, di=invariant degreesNote: for W=Sn (type An-1), Cat(W)=Cat(n) • What if W=noncrystallographic reflection group?

  15. Hyperplane arrangement • Name positive roots 1,…,m • Add affine hyperplanes defined by x, i=1 and label by I • Important in representation theory Label each 2-dim region in dominant coneby all i so that for all x in region, x, i 1= all i such that hyperplane is crossed as move out from origin 1 A2 2 3 1 2 3 3 b2 2 3 b1 1 2 2 2 3  1

  16. Regions in hyperplane arrangement Regions into which the cone x1≥x2≥…≥xnis divided by xi-xj=1, 1≤i<j≤n #6.19(lll) (Stanley, Athanasiadis, Postnikov, Shi) Regions in the dominant cone in general Ideals in the root poset

  17. Noncrystallographic case • Add affine hyperplanes defined by x, i=1 and label by i • For m even there are two orbits of hyperplanes and move one of them • When m is even roots lie on reflecting lines so symmetries break them into two orbits 1 2 4 I2(4) 3 4 a2 2 3 a1 1

  18. 12 34 12 34 12 34 2 3 4 1 2 3 2 3 4 2 3 4 1 2 3 2 3 1 2 3 2 3 2 4 2 3 2 3 2 2    Indexing dominant regions in I2(4) Label each 2-dim region by all i such that for all x in region, x, i ci= all i such that hyperplane is crossed as move out from origin These subsets of {1,2,3,4} are exactly the ideals in each case

  19. 1 4 3 1 3 2 2 1 4 3 2 1 4 3 2 Root posets and ideals I2(3) I2(4) • Express positive j in i basis • Ordering: ≤ if -═cii with ci≥0 • Connect by an edge if comparable • Increases going down • Pick any set of incomparable roots (antichain), , and form its ideal=   for all  • x, i=c  x, i /c=1 so moving hyperplane in orbit changing root length in orbit, and poset changes I2(5) 5 1 2 4 3

  20. Root poset for I2(5) Ideals index dominant regions 1 5 2 4 1 2 3 4 5 I2(5) 3 2 3 4 5 Ideals for I2(5) 1 2 3 4 1 2 3 4 5 2 3 4 5 1 2 3 4 2 3 4 3 4 2 3 3  2 3 4 5 34 2 3 3 1 4 3  2

  21. 12 34 12 34 12 34 2 3 4 1 2 3 2 3 4 2 3 4 1 2 3 2 3 1 2 3 2 3 2 4 2 3 2 3 2 2    Correspondence for m even 1 4 1 4 4 1 3 3 2 3 2 2

  22. Result for I2(m) • Theorem (Chen, K): There is a bijection between dominant regions in this hyperplane arrangement and ideals in the poset of positive roots for the root system of type I2(m) for every m.If m is even, the correspondence is maintained as one orbit of hyperplanes is dilated. • Was known for crystallographic root systems,- Shi (1997), Cellini-Papi (2002)and for certain refined counts.- Athanasiadis (2004), Panyushev (2004), Sommers (2005)

  23. Generalized Catalan numbers • Cat(I2(5))=7 but I2(5) has 8 antichains! • Except in crystallographic cases, # of antichains is notCat(I2(m)) • For any reflection group, W, Brady & Watt, Bessis define NC(W) Get |NC(W)|=Cat(W)= (h+di)/|W|, where h = Coxeter number, di=invariant degrees • But no bijection known from NC(W) to NN(W)!Open: What is a noncrystallographic nonnesting partition? • See Armstrong, Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups – will appear in Memoirs AMSand www.aimath.org/WWN/braidgroups/

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