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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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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
Outline • Partitions counted by Cat(n) • Real reflection groups • Generalized partitions counted by Cat(W) • Regions in hyperplane arrangements and the dihedral noncrystallographic case
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)
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]
P(4), NN(4), NC(4) Subposets: • NN(4)=P(4)\ • NC(4)=P(4)\
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
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)
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)
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
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)┴
3 1 2 Root poset in type A2 Root poset for A2 • Express positive j in i basis • Ordering: ≤ if -═cii 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
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
Case when n=4 Using antichains/ideals in the root poset excludes {e1-e4,e2-e3}
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?
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
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
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
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
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 -═cii 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
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
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
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)
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/