1. 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 Journal of Combinatorial Theory – Series A 32 2 2 4 4 4 3 1 3 1 1

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: P 1 ≤P 2 if P 1 refines P 2 Same as intersection lattice of H n ={x i =x j | 1≤i<j≤n} in R n under reverse inclusion Example: P(3)

4. Nonnesting partitions of [n] Nesting partition of [4]Nonnesting partition of [4] Nonnesting partitions have no nested arcs = NN(n) Examples in P(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? 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 S n Catalan number

7. Some crystallographic reflection groups Symmetries of these shapes are crystallographic reflection groups of types A 2, B 2, G 2 First two generalize to n-dim simplex and hypercube Z Corresponding groups: S n+1 =A n and S n ⋉ (Z 2 ) n = B n (Some crystallographic groups are not symmetries of regular polytopes)

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

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

10. Root System of type A 2 roots = unit vectors perpendicular to reflecting hyperplanes simple roots = basis so each root is positive or negative A2A2      e   e       e   e         e   e   i are simple roots  i are positive roots work in plane x 1 +x 2 +x 3 =0 e i -e j connect to NN(3) since hyperplane x i =x j is (e i -e j )┴

11. Root poset in type A 2 Express positive  j in  i basis Ordering:  ≤  if  -  ═  c i  i with c i ≥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  s, just write indices 1 3 2 1 (2) 3 1 (2) (2) 3 2  Root poset for A 2 Antichains (ideals) for A 2

12. NN(n) as antichains Let e 1,e 2,…,e n be an orthonormal basis of R n Subposet of intersection lattice of hyperplane arrangement {x i -x j =0 | 1≤i<j≤n} in type A n-1, { =0 | 1≤j≤n} in general Antichains (ideals) in Int(n-1) in type A n-1 (Stanley-Postnikov 6.19(bbb)), root poset in general n=3, type A 2

13. Case when n=4 Using antichains/ideals in the root poset excludes {e 1 -e 4,e 2 -e 3 }

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+d i )/|W|, where h = Coxeter number, d i =invariant degrees Note: for W=S n (type A n-1 ), Cat(W)=Cat(n) What if W=noncrystallographic reflection group?

15. Hyperplane arrangement 22  1 32 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 cone by 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 1 2 3 1 2 2 3 2 2 3 A2A2

16. Regions in hyperplane arrangement Regions into which the cone x 1 ≥x 2 ≥…≥x n is divided by x i -x j =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 When m is even roots lie on reflecting lines so symmetries break them into two orbits I 2 (4)       1 2 3 4 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

18. Indexing dominant regions in I 2 (4) Label each 2-dim region by all i such that for all x in region,  x,  i  c i = all i such that hyperplane is crossed as move out from origin   1 23 41 23 4 1 23 41 23 4 1 23 41 23 4 2 3 1 2 3 2 2 2 4 2 3 2 3 4 2 3 2 3 4 1 2 3 These subsets of {1,2,3,4} are exactly the ideals in each case

19. Root posets and ideals Express positive  j in  i basis Ordering:  ≤  if  -  ═  c i  i with c i ≥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 1 2 1 3 2 3 4 5 3 2 2 2 4 4 4 3 1 3 1 1 I 2 (3) I 2 (5) I 2 (4)

20. 1 2 3 4 5 1 2 3 4 5 2 3 4 5 1 2 3 4 2 3 4 3 4 2 3 3  Root poset for I 2 (5)Ideals index dominant regions Ideals for I 2 (5)  2 1 3 4 5 1 2 3 4 5 1 2 3 4 2 3 4 5 2 3 4 3 2 3 3 43 4 I 2 (5)

21. Correspondence for m even   1 23 41 23 4 1 23 41 23 4 1 23 41 23 4 2 3 1 2 3 2 2 2 4 2 3 2 3 4 2 3 2 3 4 1 2 3 1 1 1 3 3 3 2 2 2 444

22. Result for I 2 (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 I 2 (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(I 2 (5))=7 but I 2 (5) has 8 antichains! Except in crystallographic cases, # of antichains is not Cat(I 2 (m)) For any reflection group, W, Brady & Watt, Bessis define NC(W) Get |NC(W)|=Cat(W)=  (h+d i )/|W|, where h = Coxeter number, d i =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 AMS and www.aimath.org/WWN/braidgroups/