The combinatorics of web worlds and web diagrams

Transcription

1 The combinatorics of web worlds and web diagrams Mark Dukes University of Strathclyde, Glasgow, U.K. SLC 75, September 05 M. Dukes, E. Gardi, E. Steingrímsson, C. White. Web worlds, web-colouring matrices, and web-mixing matrices. Journal of Combinatorial Theory Series A 0 (03), no. 5, M. Dukes, E. Gardi, H. McAslan, D.J. Scott, C. White. Webs and posets. Journal of High Energy Physics 04 (04), no., -43. M.Dukes and C. White. Web matrices: structural properties and generating combinatorial identities. Preprint (05).

2 . Background and motivation Particle colliders smashing such particles together (perhaps to discover new particles) will be accompnied by a lot of quark and gluon radiation. QCD - a theory for describing quarks and gluons - the constituents of protons, neutrons and related particles. A C D B (a) Leaving aside many further com- Figure : (a) Example ments/assumptions web diagram; the (b) general Hasse diagram goal corresponding to the pose the diagram in is (a). to study the scattering amplitudes expressed as S = exp(f T RC) A Standard framework for comparing QCD to data is to calculate D scattering amplitudes related to C interaction probabilities. B (b) diagram in a given web does not necessarily have the same number of vertices. T irreducible subgraphs from one web diagram may combine to make a higher-order irr / 9

3 . Web diagrams A web diagram consists of a sequence of pegs and a set of edges, each connecting two pegs, as illustrated here: In the left diagram the indices of the pegs are shown at the bottom. The heights of the endpoints of the edges are shown in italics at each endpoint. The unique edge between pegs 3 and 6 is represented by the 4-tuple (3, 6,, 4) since the left endpoint of the edge (on peg 3) has height and the right endpoint of the edge (on peg 6) has height 4. The diagram on the right is the Feynman diagram illustration of the web diagram. / 9

4 . Web diagrams Every web diagram is uniquely represented by listing the 4-tuples that specify its edge set: D = {(,,, ), (, 7,, ), (, 4,, 3), (3, 4,, ), (3, 6,, 4), (4, 6,, 3), (4, 6, 4, ), (5, 6,, ), (5, 7,, )} The web graph G(D) of a web diagram D is the graph whose vertices represent the pegs of D, and whose labelled edges state the number of edges between pegs in D. G(D) = / 9

5 Definition. Web worlds A web world W is a set of web diagrams such that every web diagram D of W can be transformed into another web diagram D of W by permuting the vertices on pegs. Equivalently, a set of web diagrams is called a web world if G(D) = G(D ) for all D, D W. Since all diagrams in a web world have the same web graph, we can write this as G(W ). Example W = {D = {(,,, ), (,,, )}, D = {(,,, ), (,,, )}} is a web world. p p p p D D 4 / 9

6 .3 The sum of two web diagrams Suppose that we have two web diagrams D and D on the same peg set. We define the sum D D to be the web diagram that results from placing D on top of D and relabelling. Example 3 = Here D = {(, 3,, ), (3, 4,, )}, D = {(, 4,, ), (, 3,, )}, and D D = {(, 3,, ), (3, 4,, ), (, 4,, ), (, 3,, 3)}. 5 / 9

7 .4 Subweb diagrams Given a web diagram D, suppose we select a collection of edges X D. In order for X to be a web diagram, we must relabel the 3rd and 4th parts of the edges 4-tuples. Let D be our Example web diagram. Choose X = {(, 7,, ), (3, 6,, 4), (4, 6,, 3), (5, 6,, )}. 4 X = Then rel(x ) = {(, 7,, ), (3, 6,, 3), (4, 6,, ), (5, 6,, )}. 6 / 9

8 .5 Colouring and reconstructing web diagrams Suppose that D = {e i = (x i, y i, a i, b i ) : i L} is a web diagram on n pegs, and l L a positive integer. Definition 4 (Colouring and reconstruction) A colouring c of D is a surjective function c : {,..., L} {,..., l}. Let D c(j) = {e i D : c(i) = j} for all j l, the subweb diagram of D whose edges have colour j. The reconstruction Recon(D, c) W (D) of D according to the colouring c is the web diagram Recon(D, c) = rel(d c()) rel(d c()) rel(d c(l)). 7 / 9

9 .5 A colouring and reconstruction example 8 / 9

10 .5 A colouring and reconstruction example / 9

11 .5 A colouring and reconstruction example / 9

12 .5 A colouring and reconstruction example 8 / 9

13 .5 A colouring and reconstruction example 8 / 9

14 .5 A colouring and reconstruction example 8 / 9

15 .5 A colouring and reconstruction example 8 / 9

16 .5 A colouring and reconstruction example / 9

17 .5 Colouring and reconstruction Definition 5 (Self-reconstructing colourings) Let D be a web diagram and let c be an l-colouring of D. The colouring c is said to be self-reconstructing if Recon(D, c) = D. 9 / 9

18 .5 Colouring and reconstruction Definition 5 (Self-reconstructing colourings) Let D be a web diagram and let c be an l-colouring of D. The colouring c is said to be self-reconstructing if Recon(D, c) = D. Definition 6 (Colourings that produce D from D ) Given a web world W and D, D W, let F (D, D, l) = {l-colourings c of D : Recon(D, c) = D } and f (D, D, l) = F (D, D, l). 9 / 9

19 .6 Web-colouring and web-mixing matrices The following two matrices have rows and columns that are indexed by the diagrams in a given web world. The web-colouring matrix M (W ) (x) has (D, D ) entry: M (W ) D,D (x) = x l f (D, D, l). l The web-mixing matrix R (W ) has (D, D ) entry: R (W ) D,D = l ( ) l f (D, D, l), l The two are related via: 0 R (W ) D,D = M (W ) D,D (x) x dx. 0 / 9

20 Let W be the web world in Example : p p p p D D / 9

21 Let W be the web world in Example : p p p p D D There are three different colourings of D : c(e ) = c(e ) = Recon(D, c) = D c(e ) = c(e ) = Recon(D, c) = D c(e ) = c(e ) = Recon(D, c) = D / 9

22 Let W be the web world in Example : p p p p D D There are three different colourings of D : c(e ) = c(e ) = Recon(D, c) = D c(e ) = c(e ) = Recon(D, c) = D c(e ) = c(e ) = Recon(D, c) = D Consequently M (W ) D,D (x) = x and M (W ) D,D (x) = x. / 9

23 Let W be the web world in Example : p p p p D D There are three different colourings of D : c(e ) = c(e ) = Recon(D, c) = D c(e ) = c(e ) = Recon(D, c) = D c(e ) = c(e ) = Recon(D, c) = D Consequently M (W ) D,D (x) = x and M (W ) D,D (x) = x. Likewise there are three different colourings of D : c(e ) = c(e ) = Recon(D, c) = D c(e ) = c(e ) = Recon(D, c) = D c(e ) = c(e ) = Recon(D, c) = D Consequently M (W ) D,D (x) = 0 and M (W ) D,D (x) = x + x. / 9

24 Let W be the web world in Example : p p p p D D There are three different colourings of D : c(e ) = c(e ) = Recon(D, c) = D c(e ) = c(e ) = Recon(D, c) = D c(e ) = c(e ) = Recon(D, c) = D Consequently M (W ) D,D (x) = x and M (W ) D,D (x) = x. Likewise there are three different colourings of D : c(e ) = c(e ) = Recon(D, c) = D c(e ) = c(e ) = Recon(D, c) = D c(e ) = c(e ) = Recon(D, c) = D Consequently M (W ) D,D (x) = 0 and M (W ) D,D (x) = x + x. This gives ( ) ( ) M (W ) x x (x) = 0 x + x and R (W ) =. 0 0 / 9

25 .7 Web-mixing matrix example [[],[3,,],[3],[]] [[],[,,3],[3],[]] [[],[,3,],[3],[]] Let W be the web world whose web graph is G(W ) = Figure : Web whose mixing matrix is given by eq. (50). R R (W ) (,,,,) = where the diagrams are ordered as shown in figure 0. A. (,,,...,n) webs , (49) 7 5 The n = case can be obtained by relabelling external lines in the (,, ) web shown in figure, whose mixing matrix is given in eq. (47). The n = 3 case is shown in figure, and the mixing / 9

26 .8 Web-colouring and web-mixing properties The basic problems we consider are as follows: Given a web world W, What can we say about the matrices M (W ) (x) and R (W ), their entries, trace and rank? Can we determine the entries of M (W ) (x) and R (W ) for special cases? 3 / 9

27 .8 Web-colouring and web-mixing properties The basic problems we consider are as follows: Given a web world W, What can we say about the matrices M (W ) (x) and R (W ), their entries, trace and rank? Can we determine the entries of M (W ) (x) and R (W ) for special cases? Theorem 7 (Gardi & White 0) Let W be a web world. (i) The row sums of R (W ) are all zero. (ii) R (W ) is idempotent. 3 / 9

28 3. Self-reconstruction, diagonal entries, and order-preserving maps Self-reconstructing colourings Diagonal entries of M (W ) (x) Note: R (W ) idempotent trace(r (W ) ) =rank(r (W ) ) 4 / 9

29 3. Self-reconstruction, diagonal entries, and order-preserving maps Self-reconstructing colourings Diagonal entries of M (W ) (x) Note: R (W ) idempotent trace(r (W ) ) =rank(r (W ) ) Definition 8 (Decomposition poset) Let W be a web world and D W. Suppose that D = E E E k where every E i is an indecomposable web diagram. Define the partial order P = (P, ) as follows: P = (E,..., E k ) and E i E j if (a) i < j, and (b) there is an edge e = (x, y, a, b) in E i and an edge e = (x, y, a, b ) in E j such that an endpoint of e is below an endpoint of e on some peg. We call P(D) the decomposition poset of D. 4 / 9

30 Example 9 The decomposition poset P(D) we get from a web diagram D: D E P ( D) 7 E 5 E 7 E 5 E 4 E 6 E 4 E 6 E E E 3 E E E 3 Note that D = E E E 7 where E = {(,,, )} E = {(3, 4,, )} E 3 = {(5, 6,, )} E 4 = {(, 4,, ), (4, 6,, ), (4, 6, 3, )} E 5 = {(3, 6,, } E 6 = {(5, 7,, )} E 7 = {(, 7,, )} 5 / 9

31 Theorem 0 Let D be a web diagram with D = E... E k where the entries of the sum are all indecomposable web diagrams. Let P = P(D) and p = P(D). If every member of the sequence (E,..., E k ) is distinct then M (W ) D,D (x) = x +des(π) ( + x) p des(π) and R (W ) D,D = π L(P) π L(P) ( ) des(π) p ( ), p des(π) where L(P) is the Jordan-Hölder set of P (the set of all linear extensions). 6 / 9

32 Example Let D be the following web diagram: E E 3 E 3 4 Since each of the web diagrams (E, E, E 3) are distinct, Theorem 0 applies: The poset P = P(D) is the poset on {E, E, E 3} with relations E < E, E 3. We find that L(P) = {E E E 3, E E 3E }, with des(e E E 3) = 0 and des(e E 3E ) =. Consequently we have (W (n)) M D,D (x) = x( + x) + x ( + x) = x + 3x + x 3 and R (W ) D,D = ( )0 /3 + ( ) /3 ( ) = /6. 7 / 9

33 4. Web worlds having a star web graph with unitary edge weights Consider web worlds W (n) whose web graph G(W ) = star graph S n Example This web diagram D may be represented by D π where π = (5, 4,,, 6, 3) Is it possible to describe the actions of the colourings in terms of the permutations representing the diagrams? 8 / 9

34 4. Web worlds having a star web graph with unitary edge weights Consider web worlds W (n) whose web graph G(W ) = star graph S n Example This web diagram D may be represented by D π where π = (5, 4,,, 6, 3) Is it possible to describe the actions of the colourings in terms of the permutations representing the diagrams? Yes, the number of ways to colour one diagram to get another depends on the number of ways one can colour the corresponding permutation and read from it the new permutation with respect to a particular order. 8 / 9

35 4. Web worlds having a star web graph with unitary edge weights Consider web worlds W (n) whose web graph G(W ) = star graph S n Example This web diagram D may be represented by D π where π = (5, 4,,, 6, 3) Is it possible to describe the actions of the colourings in terms of the permutations representing the diagrams? Yes, the number of ways to colour one diagram to get another depends on the number of ways one can colour the corresponding permutation and read from it the new permutation with respect to a particular order. If π = (, 8, 5, 4,, 3, 7, 6) and σ = (8, 5,, 4, 3, 7,, 6) then we have Minimal(π, σ) = ((8, 5, ), (4, 3, 7), (, 6)). This means minimal(π, σ) = 3 and the unique colouring having fewest colours that transforms D π into D σ is c = (, 3,,,, 3,, ). 8 / 9

36 Theorem 3 Suppose that D π, D σ W (n) with m = minimal(π, σ). Then (W (n)) M D π,d σ (x) = x m ( + x) n m and R (W (n)) D π,d σ Consequently, ) (W (n)) R D π,d π = /n, trace (R (W (n)) = ( )m n ( ). n m = (n )! ( ) (W (n)) M D π,d π (x) = x( + x) n, trace M (W (n)) (x) = n!x( + x) n. 9 / 9

37 5. Disconnected web graphs and their connected components Let W and W be two web worlds on disjoint peg sets S and S and having web graphs G and G, respectively. 0 / 9

38 5. Disconnected web graphs and their connected components Let W and W be two web worlds on disjoint peg sets S and S and having web graphs G and G, respectively. Suppose that D, D W and D, D W. Let W 3 = W + W be a new web world which is the disjoint union of W and W. The diagram D 3 = D + D is a web diagram in W 3 and the same for D 3. Question 4 Suppose W 3 is the disjoint union of the two web worlds W and W. How can we express M W 3 D 3 (x) in terms of M W,D 3 D (x) and M W,D D?,D (x) 0 / 9

39 5. The black diamond product of power series Given A(x) = a 0 + a x a nx n and B(x) = b 0 + b x b mx m both in C[[x]] we define the black diamond product of A(x) and B(x) as A(x) B(x) = x ((( ))) k k a i b i i k 0 i,i 0, i where ((( ))) ( ) k k = i, i k i, k i, i + i k = [u i u i ](( + u )( + u ) ) k. / 9

40 5. The black diamond product of power series Definition 5 Given A () (x),..., A (m) (x) C[[x]] where A (k) (x) = n 0 a(k) n x n, we define the black diamond product of A () (x),..., A (m) (x) as: A () (x)... A (m) (x) = ((( ))) x k a () i a (m) k i m i k 0 i,...,i,..., i m m 0 where ((( k i,..., i m ))) = [u i uim m ](( + u ) ( + u m) ) k. / 9

41 5. An answer to a more general Question 4 Theorem 6 Let W,..., W m be web worlds on pairwise disjoint peg sets. Suppose that D i, D i W i for all i [, m]. Let W = W W... W m be a new web world which is the disjoint union of W,..., W m. The diagrams D = D... D m and D = D... D m are web diagrams in W and M (W ) D,D (x) = M (W ) D (x)... M (Wm),D D (x). m,d m 3 / 9

42 5.3 Disconnected web worlds and generating combinatorial identities Proposition 7 Let W be a web world that is the disjoint union of at least two web worlds. Then all entries of the web-mixing matrix R (W ) are zero, and consequently trace R (W ) = 0. 4 / 9

43 5.3 Disconnected web worlds and generating combinatorial identities Proposition 7 Let W be a web world that is the disjoint union of at least two web worlds. Then all entries of the web-mixing matrix R (W ) are zero, and consequently trace R (W ) = 0. Theorem 8 Let W be a web world whose web-colouring matrix has s different diagonal entries (H (x),..., H s(x)) that appear with multiplicities (h,..., h s). Then for all positive integers m, we have ( ) 0 a,...,as 0 a +...+as =m h a has s a=0 m a,..., a s H (x) a H s(x) The expression for the s = case is: ( ) m hh a m a m 0 H (x) a H (x) m a dx a x = 0. as dx x = 0. 4 / 9

44 5.4 Combinatorial identity example Let W be the web world we considered earlier that has web-colouring matrix ( ) M (W ) x x (x) = 0 x + x. Then H (x) = x, H (x) = x + x, h = h = and m a=0 m a k= ( i,i m a ) { }{ ( ) k+ a i!i! k i m a i }( ) k k i, k i, i + i k = 0. 5 / 9

45 6. Enumeration - number of diagrams in a web world Let W be the web world of our running example. Then Represent(W ) = Theorem 9 Let W be a web world on n pegs and A = Represent(W ). The number of different diagrams D W is W = n / (a i + a i )! i= i<j n where a i (resp. a i ) is the sum of entries in column (resp. row) i of A. a ij! 6 / 9

46 Theorem 0 7. What can we say about the square of M (W ) (x)? Let W be a web world whose diagrams each have n edges. Let D, D W and suppose that M (W ) D,D (x) = β x β nx n. Then where L i (x) = j,k x j+k ( ) M (W ) (x) = D,D b=0 a=0 n β i L i (x) i= ( )( )( j k ( ) j+k (b+a) j k b a i.e. M (W ) (x) is the image of M (W ) (x) under the operator T : C[[x]] C[[x]] which takes the basis T : (x i ) i 0 (L i (x)) i 0. n L n(x) x x 3 + x 4 3 6x 4 + x 5 + 6x 6 4 x 4 + 6x x 6 + 7x 7 + 4x 8 5 x x x x x 9 + 0x 0 6 x 5 + 6x x x x x x + 70x ab i ). 7 / 9

47 9. Repeated entries in web matrices Given a poset P = (P, ), its comparability graph comp(p) is the graph whose vertices are the elements of P, with x, y P adjacent if x y or y x. Theorem Let D and D be web diagrams in a web world W with D = E E k and D = E E k, where each of the constituent diagrams E i and E i are indecomposable. Suppose that every member of the sequence (E,..., E k ) is distinct and every member of the sequence (E,..., E k ) is also distinct. Then comp(p(d)) = comp(p(d )) = M (W ) ) D,D (x) = M(W D,D (x). 8 / 9

48 Example Let D = {(,,, ), (, 3,, ), (, 4, 3, ), (3, 5,, 3), (5, 6,, ), (5, 7,, )} and D = {(,,, ), (, 3,, ), (, 4, 3, ), (, 7,, 3), (6, 7,, ), (5, 7,, )}. The Hasse diagrams for P(D) and P(D ) are illustrated in the following diagram. Although the Hasse diagrams are clearly different, since comp(p(d)) = comp(p(d )) = G we have M (W ) ) D,D (x) = M(W D,D (x). D D P(D) P(D ) G 9 / 9