groups J. Cormier, Z. Goldenberg, J. Kelly, Plymouth State University s4 s4 s3 s1 s2 s5 Symmetric Group The symmetric group Sn is the collection of bijections from {1, 2, . . . , n} to {1, 2, . . . , n} where the operation is function composition (left ← right). Each element of Sn is called a permutation. Cycle Notation One way of representing permutations is via cycle notation. Example If σ = (1 3 5 2), then σ(1) = 3, σ(3) = 5, σ(5) = 2, σ(2) = 1, σ(4) = 4. String Diagrams A 2nd way of representing permutations is via string diagrams. Given a permutation σ, there are many ways to draw associated string diagrams. Conventions: 1. no more than 2 strings cross each other at given point, 2. strings are drawn so as to minimize crossings. Example The following string diagram corresponds to the permutation σ = (1 3 5 2). Theorem Sn is generated by the adjacent 2-cycles: s1 = (1 2), s2 = (2 3), . . . , sn−1 = (n−1 n). Relations of Symmetric Group Sn satisﬁes the following relations: 1. s2 i = 1 for all i (2-cycles have order 2) 2. short braid relations: si sj = sj si , for |i − j| ≥ 2 3. long braid relations: si sj si = sj si sj , for |i − j| = 1. Reduced Expressions If sx1 sx2 · · · sxm is an expression for σ ∈ Sn and m is minimal, then we say that the expression is reduced. By Matsumoto’s Theorem, any two reduced expressions for σ ∈ Sn diﬀer by a sequence of braid relations. Example Consider σ = s2 s1 s2 s3 s1 s2 ∈ S4 . We see that s2 s1 s2 s3 s1 s2 = s1 s2 s1 s3 s1 s2 = s1 s2 s1 s1 s3 s2 = s1 s2 s3 s2. The original expression is not reduced, but it turns out that last expression is. The only reduced expressions for σ are: s1 s2 s3 s2, s1 s3 s2 s3, s3 s1 s2 s3 . Heaps A 3rd way of representing permutations is via heaps. Fix a reduced expression sx1 sx2 · · · sxm for σ ∈ Sn . Loosely speaking, the heap for this expression is a set of lattice points (called nodes) in N × N, one for each sxi such that: 1. The node corresponding to sxi has vertical component equal to n + 1 − xi , 2. If i < j and sxi and sxj do not commute, then sxi is left of sxj . Example There are two distinct heaps for the reduced expressions from previous example: s3 s2 s1 s2 and s3 s2 s1 s3 Correspondence Between String Diagrams & Heaps There is a 1-1 correspondence between string diagrams and heaps. In the absence of a node, the string “bounces.” s4 s4 s3 s1 s2 Property T A permutation σ has Property T iﬀ there exists i such that 1. σ(i) > σ(i + 1), σ(i + 2), i i + 1 i + 2 or 2. σ(i + 2) < σ(i), σ(i + 1). i i + 1 i + 2 Example If σ = (1 3 5 2), then σ and σ−1 have Property T in 1 and 2 spots, respectively. T-avoiding σ is T-avoiding iﬀ both σ and σ−1 do not have Property T . Example The permutation σ = (1 2)(4 5) is T-avoiding. s4 s1 Theorem (Cormier, Ernst, Goldenberg, Kelly, Malbon) σ is T-avoiding iﬀ σ is a product of disjoint adjacent 2-cycles (iﬀ heap consists of a single column iﬀ string diagram consists of “bars” & “X’s”). Sketch of Proof Fix a reduced expression for σ and consider its heap. The reverse implication of the theorem is trivial. For the forward direction, consider the contrapositive. The Easy Case The easy case occurs when a node in the 2nd column in on either side is ”blocked” by at most one node in the 1st column. There are 4 possibilities: blocked 1x i i + 1 i + 2 i i + 1 i + 2 blocked 1x i i + 1 i + 2 i i + 1 i + 2 The Hard Case i i + 1 i + 2 blocked 2x apply braids up & right By applying a sequence of long braid relations, you can convert a heap in the hard case to a heap in the easy case. Coxeter Groups A Coxeter group consists of a group W together with a generating set S consisting of elements of order 2 with presentation W = S : s2 = 1, (st)m(s,t) = 1 , where m(s, t) ≥ 2 for s = t. Since s and t are elements of order 2, the relation (st)m(s,t) = 1 can be rewritten as m(s, t) = 2 =⇒ st = ts short braid relations m(s, t) = 3 =⇒ sts = tst m(s, t) = 4 =⇒ stst = tsts . . . long braid relations Coxeter groups of Types A and B The symmetric group Sn+1 with the adjacent 2-cycles as a generating set is a Coxeter group of type An . Coxeter groups of type Bn (n ≥ 2) having generating set S = {s1, s2, · · · , sn } and deﬁning relations: 1. s2 i = 1 for all i, 2. si sj = sj si if |i − j| > 1, 3. si sj si = sj si sj if |i − j| = 1 and 1 < i, j ≤ n, 4. s1 s2 s1 s2 = s2 s1 s2 s1 . Generalization of Property T to Coxeter Groups Let (W , S) be a Coxeter group and let w ∈ W . Then w has Property T iﬀ w has a reduced expression of the form stu or uts, where m(s, t) ≥ 3 and u ∈ W . In terms of heaps, w is T-avoiding iﬀ no heap for w has the property that a node in the 2nd column in on either side is ”blocked” by at most one node in outer column. Theorem (Cormier, Ernst, Goldenberg, Kelly, Malbon) In types A and B, w ∈ W is T-avoiding iﬀ w is a product of commuting generators. Concluding Remarks 1. Our advisor has classiﬁed the T-avoiding elements in type C, which consists of more than just products of commuting generators (e.g.,“sandwich stacks”). 2. Tyson Gern (University of Colorado) has classiﬁed the T-avoiding elements in type D. Again, classiﬁcation is more complicated than just products of commuting generators. 3. Our advisor is currently working with a group of students on the classiﬁcation in type F. Joint work with J. Cormier, Z. Goldenberg, J. Kelly, and C. Malbon. Research conducted under the guidance of D.C. Ernst, Plymouth State University Typeset using L ATEX, TikZ, PSTricks, and beamerposter