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Regular Polyhedra and Coxeter Groups

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Regular Polyhedra and Coxeter Groups Regular polyhedra and Coxeter groups David Vogan Introduction Linear algebra Flags Regular polyhedra and Coxeter groups Reflections Relations Classification David Vogan Face counting Tufts University, January 25, 2019 Regular polyhedra Outline and Coxeter groups David Vogan Introduction Introduction Linear algebra Ideas from linear algebra Flags Reflections Flags in polyhedra Relations Classification Face counting Reflections and relations Relations satisfied by reflection symmetries Presentation and classification Counting faces of regular polyhedra Regular polyhedra What’s the plan? and Coxeter groups David Vogan Introduction Goal: understand classification of regular polyhedra. Linear algebra Path to goal: Flags Reflections 1. Regular polyhedra ! big symmetry groups. Relations 2. Big symmetry groups Coxeter generators and relations. ! Classification Serre Analogy: matrix groups ! generators and relations. Face counting This is what you teach as Gaussian elimination. 3. So far: regular polyhedra ! finite Coxeter groups. 4. Finish: classify finite Coxeter groups. Matrix group building block:2 × 2 matrices. Coxeter group building block: Z=2Z. Regular polyhedra What’s a regular polyhedron? and Coxeter groups David Vogan Something really symmetrical. like a square Introduction Linear algebra s s Flags Reflections Relations s s Classification FIX one vertex inside one edge inside square. Face counting Two building block symmetries. ss1 s s s −! -s2 s s s s s1 takes red vertex to adj vertex along red edge; s2 takes red edge to adj edge at red vertex. Regular polyhedra More symmetries from building blocks and Coxeter . groups David Vogan r1 r identity Introduction Linear algebra r r Flags reflect in x = 0 rs1 r rs2 r reflect in y = x Reflections Relations Classification r r r r Face counting ◦ ◦ 90 rotation rs1s2 r r s2s1 r 270 rotation r r r r reflect in y = −x r s1s2s1 r r s2s1s2 r reflect in y = 0 Presentation: generators s1; s2; r r r r 2 2 ◦ relations s1 = s2 = 1, rs1s2s1s2 r 180 rotation = s s s s s1s2s1s2 = s2s1s2s1. 2 1 2 1 r r Regular polyhedra Understanding all regular polyhedra and Coxeter groups David Vogan Introduction Linear algebra Introduce a flag as a chain of faces like Flags vertex ⊂ edge in a square. Reflections Relations Introduce basic symmetries like s1, s2 which change Classification a flag as little as possible. Face counting Find a presentation of the symmetry group. See how to recover polyhedron from presentation of symmetry group. Decide which presentations are possible. Regular polyhedra Most of linear algebra and Coxeter groups David Vogan V n-diml vec space GL(V ) invertible linear maps. Introduction complete flag in V is chain of subspaces F Linear algebra Flags f0g = V0 ⊂ V1 ⊂ · · · ⊂ Vn−1 ⊂ Vn = V ; dim Vi = i: Reflections Stabilizer B(F) called Borel subgroup of GL(V ). Relations Example Classification Face counting n i V = k , Vi = f(x1;:::; xi ; 0;:::; 0) j xj 2 kg ' k . Stabilizer of this flag is upper triangular matrices. Theorem 1. GL(V ) acts transitively on flags. 2. Stabilizer of one flag is isomorphic to group of invertible upper triangular matrices. Regular polyhedra Rest of linear algebra and Coxeter groups Fix integers d = (0 = d0 < d1 < ··· < dr = n) David Vogan partial flag of type d is chain of subspaces G Introduction W0 ⊂ W1 ⊂ · · · ⊂ Vr−1 ⊂ Wr ; dim Wj = dj : Linear algebra Stabilizer P(G) is a parabolic subgroup of GL(V ). Flags Example Reflections n dj Relations V = k , Wj = f(x1;:::; xdj ; 0;:::; 0) j xi 2 kg ' k . Stabilizer is block upper triangular matrices. Classification Face counting Theorem 1. GL(V ) acts transitively on partial flags of type d. 2. Stabilizer of one flag is isomorphic to group of invertible block upper triangular matrices. 3. Consider the n − 1 partial flags obtained by omitting one proper subspace from a fixed complete flag: Gp = (V0 ⊂ · · · ⊂ Vbp ⊂ · · · ⊂ Vn) 1 ≤ p ≤ n − 1: Then GL(V ) is generated by the n − 1 parabolic subgroups P(Gp), corresponding to block upper triangular matrices with a single 2 × 2 block. Regular polyhedra Notation for polyhedra and Coxeter groups N Set C in R is convex if David Vogan X X ci 2 C; ti 2 [0; 1]; ti = 1 ) ti ci 2 C: Introduction Convex polyhedron P is intersection of half spaces Linear algebra Flags N P = fv 2 R j λi (v) ≤ ai ; 1 ≤ i ≤ Mg: Reflections N ∗ Relations Here λi 2 (R ) (dual space), ai 2 R. Classification If P is nonempty, it generates an affine subspace Face counting P S(P) = ft1q1 + ··· + tr qr j qi 2 P; ti 2 R; ti = 1g; say P is n-dimensional if S(P) is n-diml. Interior P0 of P is topological interior of P \ S(P). Boundary @P of P is P − P0. Theorem Boundary of n-diml convex polyhedron P is finite union of (n − 1)-diml convex polyhedra, the faces of P. Regular polyhedra Flags and Coxeter groups David Vogan Pn compact n-dimensional convex polyhedron Introduction F A (complete) flag in P is a chain Linear algebra P0 ⊂ P1 ⊂ · · · ⊂ Pn; dim Pi = i Flags Reflections with P − a face of P . i 1 i Relations ¨H ¨H ¨¨ HH ¨¨ HH Classification H s ¨ H s ¨ H ¨ H ¨ Face counting s H¨ s s H¨ s Two flags in two-dimls P. Symmetrys group (generated by reflections in x and y axes) is transitive on edges, not transitive on flags. Definition P regular if symmetry group acts transitively on flags. Regular polyhedra Adjacent flags and Coxeter groups David Vogan F = (P ⊂ P ⊂ · · · ⊂ P ); dim P = i 0 1 n i Introduction complete flag in n-diml compact convex polyhedron. Linear algebra Flags A flag F 0 = (P0 ⊂ P0 ⊂ · · · ⊂ P0 ) is i-adjacent to F if 0 1 n Reflections P = P0 for all j 6= i, and P 6= P0. j j i i Relations Classification @ @ @@ @@@ @ @ @@@ @@@ Face counting s s ss@@ sss @@ s s @@ss sss @ @@@@ s@ ss@@ s s ss sss Three flags adjacent to F, i = 0; 1; 2. 0 0 F 0: move vertex P0 only. F 1: move edge P1 only. 0 F 2: move face P2 only. There is exactly one F 0 i-adjacent to F (each i = 0; 1;:::; n − 1). Regular polyhedra Stabilizing a flag and Coxeter groups Lemma David Vogan Suppose F = (P0 ⊂ P1 ⊂ · · · ) complete flag in Introduction n-dimensional compact convex polyhedron Pn. Any affine Linear algebra map T preserving F acts trivially on Pn. Flags Reflections Relations Proof.Induction on n. If n = −1, Pn = ; and result is true. Classification Suppose n ≥ 0 and the the result is known for n − 1. Face counting Write pn = center of mass of Pn. Since center of mass is preserved by affine transformations, Tpn = pn. By inductive hypothesis, T acts trivially on (n − 1)-diml affine S(Pn−1) spanned by Pn−1. Easy to see that pn 2= S(Pn−1), so pn and (n − 1)-diml S(Pn−1) must generate n-diml S(Pn). Since T trivial on gens, trivial on S(Pn). Q.E.D. Compactness matters; result fails for P1 = [0; 1). Regular polyhedra Symmetries and flags and Coxeter groups David Vogan Henceforth Pn is cpt cvx reg polyhedron with fixed flag Introduction F = (P0 ⊂ P1 ⊂ · · · ⊂ Pn); dim Pi = i Linear algebra Flags Write pi = center of mass of Pi Reflections Theorem Relations There is exactly one symmetry w of Pn for each complete Classification flag G, characterized by wF = G. Face counting Corollary 0 Define Fi−1 = unique flag (i − 1)-adj to F (1 ≤ i ≤ n). There is a unique symmetrys i of Pn char by 0 si (F) = Fi−1. It satisfies 0 2 1. si (Fi−1) = F, si = 1. 2. si fixes the (n − 1)-diml hyperplane through the n points fp0;:::; pi−2; pdi−1; pi ;:::; png. Regular polyhedra Examples of basic symmetries si and Coxeter groups s −!1 David Vogan Introduction s s s s p2 p2 Linear algebra e e Flags Reflections p0 p1 p0 p1 e e ss e e ss Relations This is s1, which changes F only in P0, so acts Classification Face counting trivially on the line through p1 and p2. s p s sp s 2 2 e e -p0 p1 p0 p1 s2 e e ss e e ss This is s2, which changes F only in P1, so acts trivially on the line through p0 and p2. Regular polyhedra What’s a reflection? and Coxeter groups On vector space V (characteristic not 2), a linear David Vogan map s with s2 = 1, dim(−1 eigspace) = 1. _ Introduction −1 eigenspace is line Ls; fix basis vector α 2 V _ Linear algebra Ls = fv 2 V j sv = −vg = span(α ): Flags ∗ +1 eigspace = hyperplane Hs = kernel of nonzero α 2 V Reflections Hs = fv 2 V j sv = vg = ker(α): Relations Classification hα; vi _ sv = s _ (v) = v − 2 α : Face counting (α,α ) hα; α_i _ Extend fα g to basis of V with basis of Hs: 0−1 0 ···1 matrix of s = 0 1 ··· @ . A .. Orth reflections: quadratic form h; i identifies V ' V ∗; α = α_ ) s orthogonal. Regular polyhedra Two reflections and Coxeter groups hα ; i hα ; i s v _ t v _ David Vogan sv = v − 2 _ αs ; tv = v − 2 _ αt : hαs; αs i hαt ; αt i Introduction Assume V = Ls ⊕ Lt ⊕ (Hs \ Ht ). Linear algebra _ _ _ _ Flags On subspace Ls ⊕ Lt , basis fαs ; αt g, cst = 2hαs; αt i=hαs; αs i, Reflections −1 −c 1 0 −1 + c c c Relations s = st ; t = ; st = st ts st : 0 1 −cts −1 −cts −1 Classification Face counting det(st) = 1, tr(st) = −2 + cst cts, −1 eigenvalues exp(±i cos (−1 + cst cts=2)): Proposition Suppose −1 + cst cts=2 = real part of a prim mth root of 1, m ≥ 3; or that m = 2, and cst = cts = 0.
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