Interlacings for random walks on weighted graphs and the interchange process
aa r X i v : . [ m a t h . P R ] D ec INTERLACINGS FOR RANDOM WALKS ON WEIGHTED GRAPHSAND THE INTERCHANGE PROCESS
A. B. DIEKER
Abstract.
We study Aldous’ conjecture that the spectral gap of the interchange processon a weighted undirected graph equals the spectral gap of the random walk on this graph.We present a conjecture in the form of an inequality, and prove that this inequality impliesAldous’ conjecture by combining an interlacing result for Laplacians of random walks onweighted graphs with representation theory. We prove the conjectured inequality for severalimportant instances. As an application of the developed theory, we prove Aldous’ conjecturefor a large class of weighted graphs, which includes all wheel graphs, all graphs with fourvertices, certain nonplanar graphs, certain graphs with several weighted cycles of arbitrarylength, as well as all trees.Caputo, Liggett, and Richthammer have recently resolved Aldous’ conjecture, after inde-pendently and simultaneously discovering the key ideas developed in the present paper. Introduction
This paper studies a fundamental question arising from the theory of card shuffling, wherethe evolution of card positions is typically modeled by a Markov chain on the space of permu-tations on the set of cards. In this paper, we investigate a continuous-time Markov chain inwhich the cards at positions i and j are interchanged at rate α ij . Interchange rates may be zeroif cards at the corresponding positions cannot be interchanged. This Markov chain is knownas the interchange process . Another continuous-time Markov chain arises as the position of anarbitrary but fixed card in a deck which evolves according to the interchange process. ThisMarkov chain is known as the random-walk process .A key question is how long it takes for a deck of cards to be well-shuffled in some sense,and an important quantity in addressing questions of this form is the spectral gap . Assumingthe interchange rates α ij are chosen so that both the interchange process and the random-walk process are irreducible, the spectral gap is defined as the negative of the second largesteigenvalue of the intensity matrix of the interchange process. A conjecture of Aldous andDiaconis from 1992, often referred to as Aldous’ conjecture in the literature, says that thespectral gap of the interchange process is exactly the same as the spectral gap of the random-walk process. This conjecture, which is also listed in the open problem section of the recentbook by Levin et al. [16, Sec. 23.3], is the topic of the present paper. (Strictly speaking, theoriginal conjecture is more restrictive than in the above discussion, since it only allows α ij totake values in { , } ).It is customary to think of the interchange and random-walk processes in terms of an undi-rected, connected, weighted graph G . Each vertex of G has a label, and each edge ( i, j ) of G has an associated Poisson process with intensity α ij ≥
0. The Poisson processes on differentedges are stochastically independent. At each Poisson epoch corresponding to edge ( i, j ), thelabels at vertices i and j are interchanged. The interchange process records the positions of alllabels in the graph, while the random-walk process only records the position of a given label.Aldous’ conjecture has attracted the attention of many researchers over the past decades, butall existing results rely on some special structure on the weights α ij . These results can roughlybe categorized according to their proof methods: induction on the number of vertices in thegraph [13, 17, 22] or representation theory [4, 5, 6, 7, 10]. The current paper effectively combinesthese two approaches, and might serve as a first step towards proving Aldous’ conjecture infull generality. The main idea behind the combination of mathematical induction and representation theorycan be summarized as follows. In the induction step, a new vertex is attached to a graph forwhich it is known that the conjecture holds. If there is only one new edge incident to the newvertex, then standard eigenvalue bounds can be employed which imply that the conjectureholds for the new graph [13]. In the general case where several edges are incident to the newvertex, however, the main technical obstacle has been that the addition of this vertex maysignificantly impact the spectrum of the resulting random walk and interchange process. Thisdifficulty can be overcome with representation theory. In fact, we shall argue that the followingconjecture suitably controls the changes to the spectrum if the new vertex is of degree k − S k is the symmetric group on k letters, and( ij ) ∈ S k stands for the transposition of i and j . Conjecture 1.
Given any k ≥ , the following holds for any function g : S k → R and anynonnegative γ , . . . , γ k − : X σ ∈S k k − X i =1 γ i [ g ( σ ) − g (( ik ) σ )] ≥ X σ ∈S k X ≤ i 4. This class includes all wheel graphs, allweighted graphs with four vertices, certain graphs with weighted cycles of different lengths,certain nonplanar graphs, as well as all trees (for which the conjecture is already known tohold). It is the first time results for such general weighted graphs are obtained, illustrating thepower of knowing that Conjecture 1 holds even for small values of k .Throughout, matrix inequalities of the form A ≤ B should be interpreted as A − B beingnegative semidefinite. All vectors in this paper should be interpreted as column vectors, andwe use the symbol ⊤ for vector or matrix transpose. The ℓ × ℓ identity matrix is denoted by I ℓ . We multiply permutations from right to left, so σ ′ σ is the permutation obtained by firstapplying σ and then σ ′ .This paper is organized as follows. Sections 2–4 focus on proving that Conjecture 1 impliesAldous’ conjecture. The main technical tools are the aforementioned interlacing for Lapla-cians of random walks, which is discussed in Section 2, and a representation-theoretic viewof Conjecture 1, which is the topic of Section 3. These tools are tied together in Section 4,which contains the main argument of the proof that Conjecture 1 implies Aldous’ conjecture.Sections 5 and 6 prove Conjecture 1 in special cases: Section 5 focuses on k ≤ γ i , while Section 6 deals with general k but identical γ i . In Section 7, we present aclass of graphs for which we can prove Aldous’ conjecture using the newly developed methodol-ogy. A discussion concludes this paper, and two appendices give background on representationtheory. A postscript; independent work of Caputo, Liggett, and Richthammer. Aldous’conjecture was one of three problems targeted by an international team of researchers atthe Markov Chain Working Group in June 2009, held at Georgia Tech. I presented thispaper at that meeting, and Pietro Caputo presented a joint work with Thomas Liggett andThomas Richthammer. Both teams had posted their work on arxiv.org at the beginning ofthe meeting [3, 8]. Although the papers were written from different perspectives, we hadindependently arrived at the same proof outline: both works propose the updating rule (1)below and formulate Conjecture 1. NTERLACINGS AND THE INTERCHANGE PROCESS 3 Only days after the working group meeting, Caputo et al. were able to give a full proof ofConjecture 1. It can be found in [2]. I currently do not know if it is possible to give a differentproof of Conjecture 1 using representation theory, i.e., to complete the approach taken here.The present article is an unmodified version of my ‘working group’ preprint [8], with sometypos corrected and some arguments clarified.2. Interlacings for Laplacians of random walks on weighted graphs In this section, we state and prove an interlacing result for the weighted random-walk processon a given weighted graph G with n vertices. For other interlacing results and illustrations ofthe technique, we refer to Godsil and Royle [12, Ch. 9] or (in a slightly different setting) therecent paper by Butler [1].Let α ij ≥ i, j ) in G , i = j . We simply write α for the collection ofedge weights { α ij } . We also write { e i } for the standard basis in R n , and define w ij = e i − e j .The Laplacian of G is defined through L RWn ( α )( i, j ) = ( − α ij if i = j P j − k =1 α kj + P nk = j +1 α jk if i = j for 1 ≤ i, j ≤ n , and can thus be written in matrix form as L RWn ( α ) = X ≤ i First observe that L RWn ( α ) − L RWn ( α ′ ) = n − X i =1 α in w in w ⊤ in − X ≤ i For the last sum, we note that X ≤ i We now relate Conjecture 1 to the representation theory of the symmetric group. Back-ground on this theory is given in Appendix A. This section only contains standard resultsfrom representation theory, with a focus on transpositions of the symmetric group; see [5,Section 3] for a recent account in the context of the present paper.We write ρ λ for Young’s orthonormal irreducible representation corresponding to the parti-tion λ , and set V λij = I − ρ λij . Given the edge weights α = { α ij } of a graph G with n vertices,we set for λ ⊢ n , L λ ( α ) = X ≤ i There exists an n × n orthonormal matrix S such that for all weights α ij onthe edges of a graph with n vertices, S L RWn ( α ) S ⊤ = L ( n ) ( α ) ⊕ L ( n − , ( α ) . We note that the above decomposition of the random-walk Laplacian has a special structure.Indeed, L ( n ) ( α ) equals zero regardless of the weights α . Moreover, L ( n − , ( α ) is closely relatedto A n − reflection groups, since ρ ( n − , ij is the Householder reflection matrix corresponding tothe transposition ( ij ) of S n . That is, each ρ ( n − , ij acts on a point in R n − by reflecting it ina certain hyperplane.The Laplacian L In ( α ) of the interchange process, which is an n ! × n ! matrix, can similarlybe written as a direct sum (see also [6, Section 3E]). This Laplacian is defined through L In ( α )( σ, σ ′ ) = − α ij if σ ′ = ( ij ) σ P ≤ ℓ There exists an n ! × n ! orthonormal matrix T such that for all weights α ij onthe edges of a graph with n vertices, T L In ( α ) T ⊤ = M λ ⊢ n f λ L λ ( α ) . The preceding proposition holds without any assumption on the sign of the weights α ij .After defining signed weights on a graph with k nodes through α ij = ( γ i if j = k − γ i γ j γ + ... + γ k − if 1 ≤ i < j ≤ k − , we immediately obtain a reformulation of Conjecture 1 from Proposition 3. Lemma 4. The following is equivalent to Conjecture 1.Given any k ≥ , the following holds for any λ ⊢ k and any nonnegative γ , . . . , γ k − : k − X i =1 γ i V λik ≥ X ≤ i In this section, we prove that Conjecture 1 implies Aldous’ conjecture. We use mathematicalinduction on the number of vertices n . The conjecture trivially holds if n = 2. Suppose Aldous’conjecture holds for all graphs with n − G with n vertices, and write α ij ≥ i, j ) in G . By Proposition 2 and the fact that µ ( n )1 ( α ) = 0, Aldous’ conjecture is thatthe second smallest eigenvalue of L In ( α ) equals µ ( n − , ( α ). In view of Proposition 3, this isequivalent with µ λ ( α ) ≥ µ ( n − , ( α ) (3)for all partitions λ ⊢ n with λ = ( n ). This inequality trivially holds if λ = ( n − , G be connected; since the right-hand side of (3) vanishes if this is not the case, (3)then holds trivially since the V ij are positive semidefinite.As before, we write α ′ for the weights given by (1). The induction hypothesis yields µ ( n − , ( α ′ ) = min λ ′ ⊢ n − λ ′ =( n − µ λ ′ ( α ′ ) . To prove (3), we will show that the following string of inequalities holds if λ = ( n ) , ( n − , µ ( n − , ( α ) ≤ µ ( n − , ( α ′ ) = min λ ′ ⊢ n − λ ′ =( n − µ λ ′ ( α ′ ) ≤ µ λ ( α ) . (4)It is in the last inequality that we use Conjecture 1, but we first prove the first inequality. Lemma 5. We have µ ( n − , ( α ) ≤ µ ( n − , ( α ′ ) ≤ µ ( n − , ( α ) ≤ . . . ≤ µ ( n − , n − ( α ′ ) ≤ µ ( n − , n − ( α ) . Proof. Consider the decomposition in Proposition 2. Since L RWn ( α ) and L RWn ( α ′ ) are positivesemidefinite and L ( n ) ( α ) = L ( n ) ( α ′ ) are one-dimensional and equal to zero, we conclude that µ n ( α ) = µ n ( α ′ ) = 0 and that the largest n − L RWn ( α ) and L RWn ( α ′ ) aregiven by the eigenvalues of L ( n − , ( α ) and L ( n − , ( α ′ ), respectively. In particular, µ nk +1 ( α ) = µ ( n − , k ( α ) and µ nk +1 ( α ′ ) = µ ( n − , k ( α ′ ), and Proposition 1 yields µ ( n − , ( α ′ ) ≤ µ ( n − , ( α ) ≤ µ ( n − , ( α ′ ) ≤ . . . ≤ µ ( n − , n − ( α ′ ) ≤ µ ( n − , n − ( α ) . A. B. DIEKER The claim readily follows from these inequalities, for instance after noting that µ ( n − , ( α ′ ) = µ ( n − ( α ′ ) = 0 and µ ( n − , k ( α ′ ) = µ ( n − , k − ( α ′ ) for k ≥ (cid:3) Lemma 6. If Conjecture 1 holds, then we have for λ = ( n ) , ( n − , , min λ ′ ⊢ n − λ ′ =( n − µ λ ′ ( α ′ ) ≤ µ λ ( α ) . Proof. Under Conjecture 1, we have by Lemma 4, L λ ( α ) = X ≤ i Corollary 7. Suppose Conjecture 1 holds for k ≤ K . Let α ij ≥ be the weight of edge ( i, j ) ina given weighted graph G with n vertices, and suppose that at most K − of the n − possible α in are strictly positive.If Aldous’ conjecture holds for the graph on n − vertices induced by edge weights α ′ givenin (1), then it holds for G .Proof. The equality in (4) holds by assumption. The first inequality in (4) holds by Lemma 5,so we focus on the last inequality and the proof of Lemma 6 in particular.When at most K − n − α in are strictly positive, we may assume withoutloss of generality that these are α n , . . . , α K − ,n . By repeated application of the branching ruleand a change of basis on interchanging n and K , we get n − X i =1 α in V λin = M λ ′ ⊢ K : λ ′ ր ... ր λ K − X i =1 α in V λ ′ in ! ∼ = M λ ′ ⊢ K : λ ′ ր ... ր λ K − X i =1 α in V λ ′ iK ! , where the direct product should be taken over all possible simple paths from λ ′ to λ in theHasse diagram of Young’s lattice. Thus, one can deduce (5) from Conjecture 1 with k = K and the last inequality in (4) holds in that case. (cid:3) This corollary is of particular interest when K = 2, in which case Conjecture 1 holdstrivially. If the n -th vertex is incident to exactly one other vertex, then the modified weights α ′ on the edges of the ‘small’ graph consisting of the vertices 1 , . . . , n − α . In this context, Corollary 7 is essentially equivalent to the inductionstep in Handjani and Jungreis [13], and it readily implies that Aldous’ conjecture holds fortrees. A related argument is given by Cesi [4, Sec. 3]. NTERLACINGS AND THE INTERCHANGE PROCESS 7 Proof of Conjecture 1 for k ≤ k = 4 implies the conjecture for k < 4, so this section focuses on provingConjecture 1 for k = 4. In view of Lemma 4, it suffices to prove (2) for all λ ⊢ 4. We do so bymaking use of the explicit forms of Young’s orthonormal irreducible representations given inAppendix B. In particular, we use the vectors v λij introduced in Appendix B. Throughout thissection, for notational convenience, we suppress the superscripts λ in these vectors, so that,e.g., v ij stands for v (3 , ij in Section 5.2 while it stands for v (2 , ij in Section 5.3. We follow thesame notational convention for the superscripts λ in V λij .5.1. λ = (4) . Since V ij = 0 for 1 ≤ i < j ≤ 4, (2) trivially holds.5.2. λ = (3 , . Since v ij = v i − v j for 1 ≤ i < j ≤ 3, we readily find that( γ + γ + γ ) X i =1 γ i V i − X ≤ i 3, we find that( γ + γ + γ ) X i =1 γ i V i − X ≤ i 0, whichclearly holds. 6. Proof of Conjecture 1 for γ = . . . = γ k − This section proves Conjecture 1 for γ = . . . = γ k − . The main ingredients are Jucys-Murphy matrices and a content minimization calculation for standard Young tableaux, seeAppendix A for definitions.Fix k ≥ λ ⊢ k . We need to prove that( k − k − X i =1 V λik − X ≤ i 1) + k X i =1 c ti − kc tk , where c ti is the content of the box containing i in tableau t . The sum over i is the sum of allcontents corresponding to a Young tableau of shape λ , which can be expressed in terms of λ by noting that the sum over the contents in the j -th row equals λ j ( λ j − − ( j − λ j . Asthis is independent of t , the smallest diagonal element of the matrix on the left-hand side of(7) corresponds to a tableau t for which c tk is maximized, i.e., to a tableau t with c tk = λ − NTERLACINGS AND THE INTERCHANGE PROCESS 9 Figure 1. The graphs T ,N = K (left), T , (center), and T , (right).Therefore, we find that the smallest eigenvalue of (7) equals12 k ( k − 1) + k X i =1 c ti − k ( λ − k ( k − 1) + ∞ X j =1 λ j ( λ j − − ∞ X j =1 ( j − λ j − k ( λ − k + ∞ X j =1 λ j − ∞ X j =1 ( j − λ j − kλ = 12 ( k − λ ) + 12 ∞ X j =2 λ j − ∞ X j =1 ( j − λ j = 12 ∞ X j =2 λ j + 12 ∞ X j =2 λ j − ∞ X j =1 ( j − λ j = ∞ X j =2 λ j + ∞ X j =2 j − X i =2 λ i λ j − ∞ X j =1 ( j − λ j ≥ ∞ X j =2 λ j + ∞ X j =2 ( j − λ j − ∞ X j =1 ( j − λ j = ∞ X j =2 ( j − λ j ( λ j − . Since each λ j is a nonnegative integer, this is clearly nonnegative. This proves Conjecture 1for γ = . . . = γ k − .7. Weighted graphs with nested triangulation In this section, we introduce a class of weighted graphs for which we prove Aldous’ conjec-ture. This class includes all trees and all cycles of arbitrary length, and it arises by repeatedapplication of Corollary 7 for K = 4.Our graphs with nested triangulations are parameterized by two integers: a branching pa-rameter N ≥ D ≥ 0. For a given N , the graphs { T i,N : i ≥ } are nested in the sense that T i,N is a subgraph of T i +1 ,N for i ≥ 0. The graphs are definedrecursively as follows. Let T ,N be the complete graph on 3 vertices: T ,N = K . For eachcycle of length 3 that is present in T i,N but not in T i − ,N , we construct T i +1 ,N by adding N vertices to T i,N , and by adding 3 new edges for each new vertex to connect it to the 3 verticesof the given cycle. Thus, 3 N edges are added for each cycle of length 3 in T i,N but not in T i − ,N . The vertices of T D,N can be partitioned into D + 1 levels according to the stage atwhich they have been added. Examples are given in Figure 1. Note that T D, is a maximalplanar (triangulated) graph for any D ≥ 1, but that not all maximal planar graphs are graphswith nested triangulations. Also note that T , has K , , the complete bipartite graph on sixvertices, as a subgraph and it is therefore nonplanar by Kuratowski’s theorem. Proposition 8. For any D ≥ , N ≥ , let T D,N have arbitrary nonnegative interchangerates on its edges and assume that the graph remains connected after removing zero-rate edges.Aldous’ conjecture holds for this graph.Proof. We use induction. Since Aldous’ conjecture trivially holds for a connected graph withtwo vertices, we conclude from Corollary 7 that Aldous’ conjecture holds for the triangle K . Figure 2. The wheel graph W with 7 vertices.Since each vertex at level i +1 is incident to exactly 3 vertices at lower levels, we may repeatedlyuse Corollary 7 to deduce the claim for T i +1 ,N from the claim for T i,N . (cid:3) This proposition is of particular interest for D = N = 1, in which case T D,N is the completegraph K on four vertices. Proposition 8 then states that Aldous’ conjecture holds for all weighted graphs with four vertices.Choosing some of the interchange rates equal to zero in Proposition 8 proves Aldous’ con-jecture for some special classes of graphs. For instance, any tree can be embedded in a graphwith nested triangulations. Indeed, given any tree, let D be the maximum distance to the rootand let N be the maximum degree. It is readily seen that one can embed the tree into T D,N by mapping a vertex at distance i ≥ i in T D,N . Thus,Proposition 8 recovers the main result from [13] in this case.Instead of showing that a given graph is a subgraph of a graph with inner triangulations andappealing to Proposition 8, the following prodecure is an alternative for showing that Aldous’conjecture must hold according to the results of this paper. Corollary 7 implies that Aldous’conjecture holds for graphs which (after removing all edge weights) can be reduced to an edgeby repeatedly using the following permissible rules: • Degree-one reduction: delete a degree-one vertex and its incident edge. • Series reduction: delete a degree-two vertex k and its two incident edges ( i, k ) and( j, k ), and add in a new edge ( i, j ). • Parallel reduction: delete one of a pair of parallel edges. • Y-∆ transformation: delete a vertex k and its three incident edges ( i, k ), ( j, k ), ( ℓ, k )and add in a triangle ijℓ .These operations also appear in the context of star-triangle reducibility of a graph [9], but it isimportant to note that the ∆-Y transformation (which is the inverse of the Y-∆ transformation)is not permissible here.Wheel graphs are examples of graphs which can be reduced to an edge using these operations.We write W n for the wheel graph with n vertices, see Figure 2 for W . Indeed, one obtains W n − from W n after applying a Y-∆ transformation to one of the outer vertices of W n followedby three parallel reductions. This procedure can be repeated until W = K arises, which isreadily reduced to an edge. Note that cycles are subgraphs of wheel graphs: choose theinterchange rates on the spokes of the wheel equal to zero except for two adjacent spokes, andalso let the interchange rate vanish on the edge incident to the two outer vertices of these twospokes. Thus, we have also proven that Aldous’ conjecture holds for weighted cycles.8. Discussion Other Markov processes with the same spectral gap as the random walk. Apartfrom the interchange process and the random walk, several other natural Markov chains arisefrom the interchange dynamics on a weighted graph. Indeed, one may allow several verticesto receive the same label, which can be thought of as a color. Interchanging nodes with thesame color then does not change the color configuration on the graph. Thus, for each possibleinitial configuration of colors, one obtains a continuous-time Markov chain. One can think of NTERLACINGS AND THE INTERCHANGE PROCESS 11 these processes as parameterized by Young diagrams (partitions), where each row correspondsto a color and the number of boxes in each row correspond to the number of vertices to receivethis color. The resulting process can be interpreted as a random walk on a so-called Schreiergraph, see also Cesi [4]. The interchange process is a special case of this construction with λ = (1 n ), i.e., all vertices have different colors. Similarly, the random walk process arises onsetting λ = ( n − , λ naturally arises from the M λ module in representation theory. The multiplicities of the irreducible representations are givenby the so-called Kostka numbers. As a consequence of the resulting block structure, all of theintensity matrices (except for the trivial one corresponding to λ = ( n )) contain the irreduciblerepresentation corresponding to the partition ( n − , Gelfand-Tsetlin patterns. By Proposition 1, subsequent removal of vertices and updatingof the weights according to (1) yields a Gelfand-Tsetlin pattern, i.e., a collection of subse-quent interlaced sequences. Subsequent removal of vertices without weight updating yields anondecreasing spectral-gap sequence, an observation which has previously proven useful in thecontext of Aldous’ conjecture [17, 22]. The significance of the Gelfand-Tsetlin structure iscurrently unclear. The cut-off phenomenon. It is a natural question whether the results of this paper canbe exploited to study the cut-off phenomenon for Markov chains. This question is currentlyopen. Proving a cut-off phenomenon requires control over the whole spectrum, not only nearthe edge. A variety of known results [20], e.g., on ℓ -adjacent transposition walks, suggests that(pre)cut-off thresholds for interchange processes have an extra log( n ) factor when comparedto the corresponding random walk processes. Proposition 3 suggests that the second smallesteigenvalue of the Laplacian of the interchange process typically has multiplicity f ( n − , = n − n ) factor. Electric networks. Section 7 showed how the Y-∆ transformation naturally arises in thecontext of Corollary 7, but there may be a deeper connection. For n = 4, the definition of α ′ in (1) in terms of α appears in formulas for the resistance in electrical networks when a ∆ istransformed into a Y. The recent work of Caputo et al. [2] sheds some light on this. Acknowledgments The author would like to thank Prasad Tetali and David Goldberg for valuable discussions,and Kavita Ramanan for helpful comments on an earlier draft. Appendix A. Background on representation theory This section reviews the elements of representation theory used in the body of this paper.More comprehensive accounts can be found in [19, 11, 15, 18].A partition λ of n , written λ ⊢ n , is a sequence of nonnegative integers ( λ , λ , . . . ) with P ∞ j =1 λ j = n and λ ≥ λ ≥ . . . . For notational convenience, we suppress the zero elementsof the sequence. Also, if the integer k appears m times in the partition λ , we replace the m copies of k by a single copy of k m . For instance, (4 , , 1) is shorthand for (4 , , , , , . . . ). Apartition can be identified with a Young diagram, which is a collection of n boxes arranged inleft-justified rows, with the i -th row containing λ i boxes. For instance,(4 , , 1) = . Given two partitions λ ′ and λ , we write λ ′ ր λ if the Young diagram of λ can be obtainedfrom the Young diagram of λ ′ by adding a box. For instance, we have ր , since a box is added to the second row. A natural related partial order on partitions is definedby diagram containment. The set of all partitions equipped with this partial order is called Young’s lattice .Given a partition λ , a standard Young tableau with shape λ ⊢ n is the Young diagramcorresponding to λ with each of the numbers 1 , . . . , n inside one of the n boxes, in such a waythat the numbers in each of the rows as well as in each of the columns of the Young diagramare increasing. For instance, 1 2 3 45 67 and 1 3 5 72 46are both standard Young tableaux with shape (4 , , f λ for the number of differentstandard Young tableau with shape λ . Note that a Young tableau with shape λ can alterna-tively be thought of as a saturated chain in Young’s lattice starting from the empty partition[21, Prop. 7.10.3].The content of a box in a Young diagram is defined as the x -coordinate of the box minusits y -coordinate. Thus, the boxes in the following Young diagram contain their content:0 1 2 3 − − . For instance, in the two given examples of standard Young tableaux, the box containing 7 hascontent − c ti for the content of the box containing i in tableau t . We next introduce Young’s orthonormal irreducible representation corresponding to thepartition λ ⊢ n . This is a family of f λ × f λ matrices { ρ λσ : σ ∈ S n } parameterized by elementsof S n , such that the matrices behave in the same way as the elements of S n when multipliedtogether (i.e., the mapping σ ρ λσ is a group homomorphism). When σ is a transposition,say σ = ( ij ), we write ρ λij instead of ρ λ ( ij ) . We let the standard basis vectors correspond to thestandard Young tableaux with shape λ under the dictionary (total) order, i.e., if the numbersare read from left to right by rows, starting at the top row, the first digit in which two tableauxdisagree will be larger for the larger tableau. We fix 2 ≤ i ≤ n and specify ρ λi − ,i ; since theadjacent transpositions generate S n , this specifies the whole group representation: • If i and i + 1 are in the same row of t , then ρ λi,i +1 ( t, t ) = 1. • If i and i + 1 are in the same column of t , then ρ λi,i +1 ( t, t ) = − • Suppose i and i + 1 are not in the same row or column of t . Write s for the standardYoung tableau resulting from swapping i and i + 1 in t . Then we have (cid:18) ρ λi,i +1 ( t, t ) ρ λi,i +1 ( t, s ) ρ λi,i +1 ( s, t ) ρ λi,i +1 ( s, s ) (cid:19) = (cid:18) r − √ − r − √ − r − − r − (cid:19) , where r = c ti +1 − c ti , the axial distance between the boxes containing i and i + 1.The elements of ρ λi,i +1 left unspecified by these three rules are zero. A branching rule holds,which reduces to ρ λij = M λ ′ ⊢ n − λ ′ ր λ ρ λ ′ ij NTERLACINGS AND THE INTERCHANGE PROCESS 13 for transpositions ( ij ) with i < j < n .Of special importance in representation theory are the so-called Jucys-Murphy elements ;they play a key role in the Vershik-Okounov approach to representation theory [18]. For 2 ≤ j ≤ f λ , the Jucys-Murphy matrix corresponding to the partition λ is defined as X λj = P j − i =1 ρ λij .Their significance stems from the fact that these matrices commute; in fact, they are diagonalmatrices. Element ( t, t ) of X λj equals c tj , the content of the box containing j in tableau t . Appendix B. Young’s orthonormal irreducible representations of S This appendix evaluates Young’s orthonormal irreducible representations of S at transpo-sitions, which is a key ingredient in Section 5. The given formulas can be verified with thedefinition of Young’s orthonormal irreducible representation in Appendix A.For 1 ≤ i < j ≤ 4, we have ρ (4) ij = 1 (one-dimensional).For 1 ≤ i < j ≤ 4, we have ρ (3 , ij = I − v (3 , ij v (3 , ij ⊤ , where v (3 , = √ , v (3 , = p / p / , v (3 , = p / p / p / ,v (3 , = p / − p / , v (3 , = p / p / − p / , v (3 , = p / − p / . For 1 ≤ i < j ≤ 4, we have ρ (2 , ij = I − v (2 , ij v (2 , ij ⊤ , where v (2 , = (cid:18) √ (cid:19) , v (2 , = (cid:18) p / p / (cid:19) , v (2 , = (cid:18) p / − p / (cid:19) ,v (2 , = (cid:18) p / − p / (cid:19) , v (2 , = (cid:18) p / p / (cid:19) , v (2 , = (cid:18) √ (cid:19) . For 1 ≤ i < j ≤ 4, we have ρ (2 , ) ij = − I + v (2 , ) ij v (2 , ) ij ⊤ , where v (3 , = √ , v (3 , = p / − p / , v (3 , = p / − p / p / ,v (3 , = − p / − p / , v (3 , = − p / − p / p / , v (3 , = p / p / . For 1 ≤ i < j ≤ 4, we have ρ (1 ) ij = − References [1] S. 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