The White House This photo shows the South side of the Mansion; visitors tour the the ’South Lawn’. To the west (left of photo) is the Old Executive Office Building (not shown); to the south is the National Mall (also not shown). Photo Courtesy of the Washington, D.C. Convention & Visitor Center go to the White House page -orgo to main tour page Sarah Whitehouse, -(Co)homology of commutative algebras and some related representations of the symmetric group, Ph.D. thesis, Warwick University, 1994. 2 The module Lien Let V be a complex vector space with basis x ; : : : ; xn. Let Lien be the part of the free Lie algebra L(V ) that is of degree one in each xi. 1 dim Lien = (n 1)! Basis: [ [[x ; xw ]; xw ]; : : : ; xw n ]; where w permutes 2; 3; : : : ; n. 1 (2) 3 (3) ( ) The symmetric group Sn acts on Lien by permuting variables. (1; 2) [[x = [[x 1 ; x3]; x2] 2 3 1 1 ; x2]; x3 1 3 2 4 = [[x ; x ]; x ] ] + [[x ; x ]; x ] For any function f : Sn ! C , recall that 1 X ch f = f (w )p w ; n! w2Sn where if w has i i-cycles then 1 2 p w = p p ; ( ( P ) 1 ) 2 with pk = i xki . In particular, if is the irreducible character of Sn indexed by ` n, then ch = s; the Schur function indexed by . 5 = subgroup of Sn generated by (1; 2; : : : ; n) Cn Theorem. As an Sn-module, n e i=n: Lien = indS Cn Hence 1X n=d (d)pd : ch(Lien) = n djn 2 6 Theorem. Let be the irreducible character of Sn indexed by ` n. Then hLien; i = #SYT T of shape ; maj(T ) 1 (mod n); where X maj(T ) = i: i+1 1 3 4 2 below i 1 2 3 4 ch Lie = s + s 4 31 7 211 Other occurrences of Lien n = lattice of partitions of [n], ordered by renement ~ i(n) H = ith (reduced) homology group (over Q , say) of (order complex of) n As Sn-modules, 0 ; i 6= n 3 ~ Hi(n) = sgn Lien; i = n 3: 8 1234 123 12 124 13 134 234 23 14 12-34 13-24 34 24 Π4 9 14-23 Tn = set of rooted trees 0 with endpoints labelled 1; 2; : : : ; n; and no vertex with exactly one child Schroder (1870) showed (the fourth of his vier combinatorische Probleme ) that X xn #Tn = (1 + 2x ex)h i ; n! n where F (F h i) = F h i(F (x)) = x. For T; T 0 2 Tn , dene T T 0 if T can be obtained from T 0 by contracting internal edges. Note: 1 0 1 1 1 0 10 2 3 1 3 1 1 3 2 1 2 3 Theorem. As Sn-modules, 0 ; i 6= n 3 ~ Hi(Tn ) = sgn Lien; i = n 3: 0 11 2 A \hidden" action of Sn on Lien (Kontsevich) 1 Let Ln be the free Lie algebra on n generators x ; : : : ; xn, and let h ; i = nondegenerate inner product on Ln satisfying h[a; b]; ci = ha; [b; c]i: For ` 2 Lien and w 2 Sn, let h`; xniw = h`w ; xw n i ! h`0; xni: 1 1 straighten ( ) So the map w : Lien ! Lien dened by w(`) = `0 denes an Sn action on Lien , the Whitehouse module Wn for Sn or the cyclic Lie operad. (Explicit description of action of (n 1; n) by H. Barcelo.) 1 1 12 1 dim Wn = dim Lien Wn 1 = indSSn Lien n 1 ch Wn = p1 n X 1 dj n 1X ( n djn = (n 2)! 1 Lien: n (d)pd ( =d 1) 1) n=d (d)pd hWn; i = #SYT T of shape , maj(T ) 1 (mod n 1) #SYT T of shape , maj(T ) 1 (mod n) (Not a priori clear that this is 0.) 13 s2; s111; s22; s311 s42 + s3111 + s222 s511 + s421 + s331 + s3211 + s22111 + 2s + 422 Getzler-Kapranov: Mn ; = Lien; where M is the irreducible Sn-module indexed by . Wn 1 1 14 Other occurrences of Wn Nonmodular partitions (Sundaram) n = f 2 n : has at least two nonsingleton blocksg Theorem. As Sn-modules, 0 ; i 6= n 4 ~ Hi(n) = sgn Wn; i = n 4: 15 123 45 124 134 35 25 234 125 135 15 34 24 235 145 245 14 23 13 345 12 25 24 23 15 14 13 15 14 12 15 13 12 14 13 12 34 35 45 34 35 45 24 25 45 23 25 35 23 24 34 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 Σ5 16 Homeomorphically irreducible trees (Hanlon, after Robinson-Whitehouse) Tn = set of free (unrooted) trees with endpoints labelled 1; 2; : : : ; n, and no vertex of degree two For T; T 0 2 Tn, dene T T 0 if T can be obtained from T 0 by contracting internal edges. 17 1 3 1 2 1 2 2 4 3 4 4 3 1 2 3 4 Theorem. As Sn-modules, 0 ; i 6= n 4 ~ i(Tn) H = sgn W ; i = n 4: n 18 Partitions with block size at most k, (n 1)=2 k n 2 (Sundaram) n;k = poset of partitions of f1; : : : ; ng with block size at most k Assume (n 1)=2 k n 2. 19 12 14-23 13-24 12-34 34 13 24 14 23 Π 4,<2 Note: ~ i( ; ; Z ) H = 4 2 20 0; i 6= 0 Z ; i = 0: 2 Theorem (Sundaram): 0; i 6= n 4 Z n ; i = n 4: Moreover, as Sn-modules, ~ n (n;k ; Q ) H = sgn Wn: ~ i(n;k ; Z ) H = ( 4 21 2)! Not 2-connected graphs (Babson et al., Turchin) = loopless graph without multiple edges on the vertex set f1; : : : ; ng. Identify G with its set of edges. G is 2-connected if it is connected, and removing any vertex keeps it connected. G n = simplicial complex of not 2connected graphs on 1; : : : ; n. 22 ~ 1(3; Z ) H 23 = Z Theorem (Babson-Bjorner-Linusson-ShareshianWelker, Turchin): 0; i 6= 2n 5 ~ Hi(n; Z ) = Z n ; i = 2n 5 ( 2)! Moreover, as Sn-modules, ~ n (n; Q ) H = Wn: 2 5 24 General technique: acts on ; w = fF 2 : G w 2G = Fg w F Hopf trace formula =) X w ~ i()) ~( ) = ( 1)i tr( w; H | {z } character value at w of G acting on H~ i() Use topological or combinatorial techniques such as lexicographic shellability to show that H~ i() vanishes except for one value of i. 25 Explicit Sn-equivariant isomorphisms (up to sign) between the cyclic Lie operad, the cohomology of the tree complex Tn, and the cohomology of the complex n of not 2-connected graphs were constructed by M. Wachs. Note: 26 A q-analogue of a trivial Sn-action (Hanlon-Stanley) For w 2 Sn and q 2 C , let `(w ) = #inversions of w X `w q w 2 C Sn n (q ) = w2Sn n(q ) acts on C Sn by left multiplication. Theorem (Zagier, Varchenko): ( det n(q) = n Y k =2 1 27 ) q k (k 1) n !( n k +1)=k (k 1) Proof (sketch). Let Tn ( q ) Easy: = n X qj 1 j =1 (n; n 1; : : : ; n j +1): n(q ) = T2(q )T3(q ) Tn ( q ) : Let a b and [a; b] = (a; a 1; : : : ; b) 2 Sn: Let Gn = (1 qn[n 1; 1])(1 qn [n 1; 2]) (1 1 (1 qn = [n; 1])(1 Hn 1 1 Duchamps et al.: qn Tn 28 2 q 2) [n; 2]) (1 q[n; n 1]): = GnHn: Theorem. Let = e i=n n as Sn-modules we have ker n( ) = Wn: 2 29 ( 1) . Then Transparencies available at: http://www-math.mit.edu/ rstan/trans.html 30 REFERENCES 1. E. Babson, A. Bjorner, S. Linusson, J. Shareshian, and V. Welker, Complexes of not i-connected graphs, MSRI preprint No. 1997-054, 31 pp. 2. G. Denham, Hanlon and Stanley's conjecture and the Milnor gre of a braid arrangement, preprint. 3. G. Duchamps, A. A. Klyachko, D. Krob, and J.-Y. Thibon, Noncommutative symmetric functions III: Deformations of Cauchy and convolution algebras, preprint. 4. E. Getzler and M. M. 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