ON EIGENFUNCTIONS OF THE STAR GRAPHS Vladislav Kabanov Institute - - PowerPoint PPT Presentation

ON EIGENFUNCTIONS OF THE STAR GRAPHS

Vladislav Kabanov Institute of Mathematics and Mechanics Yekaterinburg, Russia Joint work with

• S. Goryainov, E. Konstantinova,
• L. Shalaginov, and A. Valyuzhenich

Shanghai Jao Tong University, March 14, 2019

On eigenfunctions of the Star Graphs

Outline

• Eigenfunctions of graphs
• Cayley graph Cay(G, S) and Star graph Sn
• PI-eigenfunctions of the Star graph Sn
• A generalization of PI-eigenfunctions
• Partitions and Young tableaux
• Polytabloid eigenfunctions
• Connection between polytabloid eigenfunctions and

PI-eigenfunctions

On eigenfunctions of the Star Graphs

Eigenfunctions of a graph

Let Γ = (V, E) be a graph. A function f : V − → R is called an eigenfunction of the graph Γ corresponding to an eigenvalue θ if f ≡ 0 and for any vertex x the local condition θ · f(x) =

• y∈Γ(x)

f(y) holds, where Γ(x) is the neighbourhood of x.

On eigenfunctions of the Star Graphs

Cayley graph and the neighbourhood of a vertex

Let G be a finite group and S be a subset of G which does not contain the identity element and is closed under inversion. The Cayley graph of G generated by S, denoted by Cay(G, S), is the graph with:

• vertex set G,
• edge set {(x, sx)| for each x ∈ G, s ∈ S}.

By the choice S the Cayley graph Cay(G, S) is an undirected graph without loops and multiple edges.

On eigenfunctions of the Star Graphs

Cayley graph and the neighbourhood of a vertex

Let Γ be a Cayley graph Cay(G, S). For a given vertex x in Γ, let Γ(x) = Sx = {sx|s ∈ S}. Then Γ(x) is the neighbourhood of x in Γ.

On eigenfunctions of the Star Graphs

Group algebra C[G] and the action by multiplication

Let C be a complex field and let C[G] be the group algebra of G

• ver C.

For a subset S in G, consider the element S ∈ C[G] given by S =

• s∈S

s. Left multiplication of elements from C[G] by S is a linear transformation of C[G]. It is known the matrix of this linear transformation coincides with the adjacency matrix of Cay(G, S), hence spectral theory

• f Cayley graphs is connected with the theory of group

representations.

On eigenfunctions of the Star Graphs

The Star graph Sn

Let Ω be the set {1, . . . , n}, n ≥ 2. We consider the symmetric group SymΩ and put S = {(1 2), (1 3), . . . , (1 n)}. The Star graph Sn is the Cayley graph over the symmetric group SymΩ with the generating set S.

On eigenfunctions of the Star Graphs

The Star graph Sn

The Star graph is interesting as network topology, because it is an attractive alternative to the hypercube, a popular network for interconnecting processors in a parallel computer, and it compares favorably with hypercube in several aspects. In this talk we are interested in the eigenfunctions and spectrum of the Star graph Sn.

• R. Ko’tter and F. R. Kschischang. Coding for errors and

erasures in random network coding. IEEE Trans. Inf. Theory, 54, 8 (2008) 3579–3591.

On eigenfunctions of the Star Graphs

The spectrum of the Star graph Sn

It was shown by Krakovski and Mohar that the spectrum of the Star graph Sn contains all integers from −n + 1 to n − 1 (except 0 if n = 2 or n = 3). Since the Star graph is bipartite, mul(n − k) = mul(−n + k) for each integer 1 k n. Moreover, ±(n − 1) are simple eigenvalues of Sn.

• R. Krakovski, B. Mohar, Spectrum of Cayley graphs on the

symmetric group generated by transposition, Linear Algebra and its Applications, 437 (2012) 1033–1039.

On eigenfunctions of the Star Graphs

The spectrum of the Star graph Sn

In [1], Chapuy and Feray pointed out that the spectrum of the Star graph Sn is integral, which was actually shown by Jucys in [2], who considered the action of the element (n 1) + (n 2) + . . . + (n n − 1)

• n the complex group algebra over symmetric group C[Symn]

by left multiplication. However, eigenfunctions corresponding to these integral eigenvalues were never obtained explicitly. [1] G. Chapuy, V. Feray, A note on a Cayley graph of Symn, arXiv:1202.4976v2 (2012) 1–3. [2] A. Jucys, Symmetric polynomials and the center of the symmetric group ring, Reports Math. Phys., 5 (1974) 107–112.

On eigenfunctions of the Star Graphs

PI-eigenfunctions of the Star graph Sn

In this talk, we present a family of eigenfunctions of Sn, n 3 for any eigenvalue n − m − 1, n > 2m.

• S. Goryainov, V. V. Kabanov, E. Konstantinova, L. Shalaginov,
• A. Valyuzhenich, PI-eigenfunctions of the Star graphs,

arXiv:1802.06611 (2018) 1–19. Submitted to Linear Algebra and its Applications

On eigenfunctions of the Star Graphs

PI-eigenfunctions of the Star graph Sn

Let us define a m-tuple Im = (i1, i2, . . . , im) of m pairwise distinct elements from the set {1, . . . , n} and a m-tuple Pm = ((j1, k1), (j2, k2), . . . , (jm, km)), where 2m pairwise distinct elements from the set {2, . . . , n} are arranged into m pairs. Define a function fPm

Im : Symn → R. For a permutation

π = [π1π2 . . . πn] ∈ Symn, we put fPm

Im (π) = 0, if there exists

t ∈ {1, 2, . . . , m} such that πjt = it and πkt = it. If for every t ∈ {1, 2, . . . , m} either πjt = it or πkt = it, then we define a binary vector Xπ = (x1, x2, . . . , xm) as follows: xt = 1, if πjt = it; 0, if πkt = it. We use the vector Xπ to complete the definition of the function fPm

Im :

On eigenfunctions of the Star Graphs

PI-eigenfunctions of the Star graph Sn

fPm

Im (π) =

   1, if wt(Xπ) is an even number; −1, if wt(Xπ) is an odd number; 0, if there exists t such that πjt = it and πkt = it.

On eigenfunctions of the Star Graphs

PI-eigenfunctions of the Star graph Sn

fPm

Im (π) =

   1, if wt(Xπ) is an even number; −1, if wt(Xπ) is an odd number; 0, if there exists t such that πjt = it and πkt = it. Theorem 1. For n 3 and n > 2m > 0 the function fPm

Im is an

eigenfunction with eigenvalue n − m − 1 of the Star graph Sn. We call the eigenfunction from Theorem 1 PI-eigenfunctions.

On eigenfunctions of the Star Graphs

A generalization of PI-eigenfunctions

Further we present a generalization of the family of PI-eigenfunctions of the Star graph Sn, n 3, for all its non-zero eigenvalues. To present the generalized construction of eigenfunctions, we introduce some new definitions.

On eigenfunctions of the Star Graphs

A generalization of PI-eigenfunctions

Let us define two subsets Λ and ∆ of Ω with their partitions as follows. For any positive integers m and c such that m < n − 1 and c < n − m − 1, let Λ be an m-element subset of Ω and Λ1, . . . , Λc be its partition with parts of cardinality λ1, . . . , λc,

• respectively. Thus, m = λ1 + . . . + λc. Denote this partition of m

by λ = (λ1, . . . , λc). Let ∆ be an (m + c)-element subset of Ω \ {1} and ∆1, . . . , ∆c be its partition with parts of cardinality δ1, . . . , δc, respectively, where δt = λt + 1 for each t ∈ {1, . . . , c}. Thus, m + c = δ1 + . . . + δc. Denote this partition of m + c by δ = (δ1, . . . , δc).

On eigenfunctions of the Star Graphs

A generalization of PI-eigenfunctions

Let V1 and V2 be m-ary and c-ary Cartesian powers of Ω, correspondingly, and V = V1 × V2 be their Cartesian product. For given Λ and ∆, we take tuples IΛ ∈ V1, P∆ ∈ V each of which contains only pairwise distinct elements of Ω. The m-tuple IΛ is given by the ordered parts (I1, . . . , Ic) with respect to the partition λ of m such that It = (it1, . . . , itλt), where its ∈ Λt, 1 s λt, for every t, 1 t c. The (m + c)-tuple P∆ is given by the ordered parts (P1, . . . , Pc) with respect to the partition δ of m + c such that Pt = (jt1, . . . , jtδt), where jts ∈ ∆t, 1 s δt, for every t, 1 t c.

On eigenfunctions of the Star Graphs

A generalization of PI-eigenfunctions

Let us consider the subgroup G of SymΩ that fixes all symbols from Ω \ ∆. Let H = {τ ∈ SymΩ : τ(∆t) = ∆t, 1 t c} be the stabilizer of the partition ∆1, . . . , ∆c, where τ(∆t) = ∆t means that τ(j) ∈ ∆t for any j ∈ ∆t. For any π ∈ SymΩ and τ ∈ H, by the relation IΛ ≺ π(τ(P∆)), where IΛ ∈ V1 and π(τ(P∆)) ∈ V , we mean that all entries of π(τ(P∆)) are the same as entries of IΛ at the corresponding positions, and π(τ(jtλt)) / ∈ Λt.

On eigenfunctions of the Star Graphs

A generalization of PI-eigenfunctions

We define the following function: fP∆

IΛ (π) =

   +1, if IΛ ≺ π(τ(P∆)) for even permutation τ ∈ H; −1, if IΛ ≺ π(τ(P∆)) for odd permutation τ ∈ H; 0, if IΛ ≺ π(τ(P∆)) for any permutation τ ∈ H. (1)

On eigenfunctions of the Star Graphs

A generalization of PI-eigenfunctions

fP∆

IΛ (π) =

   +1, if IΛ ≺ π(τ(P∆)) for even permutation τ ∈ H; −1, if IΛ ≺ π(τ(P∆)) for odd permutation τ ∈ H; 0, if IΛ ≺ π(τ(P∆)) for any permutation τ ∈ H. (2) Theorem 2. Let n 3 and n − m − 1 > 0, then the Star graph Sn has the eigenfunction fP∆

IΛ

with eigenvalue n − m − 1.

On eigenfunctions of the Star Graphs

The eigenfunction of S4 for eigenvalue 1

On eigenfunctions of the Star Graphs

Proof of Theorem 2

We put I = IΛ, P = P∆, and f = fP∆

IΛ . To show that f is an

eigenfunction, it is enough to verify that the local condition holds for any permutation π = [π1 . . . πn] ∈ SymΩ. Suppose π1 = its for some its ∈ I. Since 1 / ∈ P, we have f(π) = 0 by (1). If the set of neighbours of π with non-zero value of f is not empty, then there exist a permutation τ ∈ H such that I ≺ ((1 jts) ◦ π)(τ(P)), and hence f((1 jts) ◦ π) = 0. But in this case there is the only symbol jts‘ in ((1 jts) ◦ π)(τ(P)) \ I such that f((1 jts′) ◦ π) = 0. Moreover, (its its′) is an odd permutation from H, hence −f((1 js′) ◦ π) = f((1 js) ◦ π), which means that the local condition holds in this case.

On eigenfunctions of the Star Graphs

Proof of Theorem 2

Suppose π1 = its for each t and s such that 1 t c and 1 s δt. If f(π) = 0, then π has no neighbours with non-zero value of f and local condition holds. If f(π) = 0, then the neighbours of π with non-zero value of f form the set R = {(1 r) ◦ π | r ∈ {2, . . . , n}, πr ∈ I}. The cardinality of this set equals n − m − 1. Since f(σ) = f(π) for any σ ∈ R, then

r∈R

f((1 r) ◦ π) = n − m − 1. Thus, local condition holds for the permutation π in this case, which completes the proof.

On eigenfunctions of the Star Graphs

Partitions and Young tableaux

Let n = n1 + n2 + · · · + ns be a partition of integer n. It is convenient to display this partition as a diagram. For example, take the partition λ = (4, 2, 1) of the number n = 7. Then the corresponding diagram is as follows: If all cells in the diagram are labeled of integers {1, . . . , n}, then we have got a tableau. A tableau t is standard if of all cells in the diagram are labeled increasing sequence of integers. For i ∈ {1, . . . , n} we define ct(i) = x − y if i in the column x and the row y. So for example, for the next standard tableau we have the value ct(7) = 4 − 1 = 3. 1 2 6 7 3 5 4

On eigenfunctions of the Star Graphs

Decomposition of the regular representation of Symn

The regular representation of Symn is decomposed into irreducible submodules as follows C[Symn] =

• λ∈P(n)

mλVλ, where P(n) the set of partitions of n and mλ = dimVλ.

On eigenfunctions of the Star Graphs

A result by Jucys

Jucys Theorem. (1974) Let λ ∈ P(n). Then there exists a basis {vt} of the irreducible module Vλ, indexed by standard Young tableaux t of shape λ such that for all i ∈ {2, . . . , n}, the following equality Jivt = ct(i)vt holds. If i = n, the theorem says that there exists a basis of an irreducible module Vλ consisting of eigenvectors of Jn. Moreover, the number of independent eigenvectors corresponding to the same eigenvalue is given by the number of standard Young tableaux of the shape λ with the same value ct(n).

On eigenfunctions of the Star Graphs

The spectrum of the Star graph Sn

• Corollary. For any n ≥ 4, the spectrum of the Star graph Sn

consists of integers −(n − 1), . . . , n − 1, and the multiplicity of an eigenvalue θ is given by the formula mul(θ) =

• λ∈P(n)

mλ · Cλ(k), where mλ is the number of standard Young tableaux of shape λ and Cλ(k) is the number of standard Young tableaux of shape λ with c(n) = k.

On eigenfunctions of the Star Graphs

Permutation module M λ

Let λ be a shape with n cells. For a tableau t of shape λ, the λ-tabloid {t} is the set of all tableaux of shape λ that can be obtained from t by permutations of elements in rows. Let Mλ = C{{t1}, . . . , {tk}} be the permutation module corresponding to λ, where {t1}, . . . , {tk} is a complete list of λ-tabloids. Further, we consider the action of the group algebra C[Symn] on Mλ.

On eigenfunctions of the Star Graphs

Polytabloids

Let t be a tableau of shape λ. Let Ct be the column stabilizer of Ct. Put et :=

• π∈Ct

sgn(π){π(t)}. The element et ∈ Mλ is called the polytabloid given by the tableau t.

On eigenfunctions of the Star Graphs

Specht module Sλ and its standard basis

Given a partition λ, the corresponding Specht module Sλ, is the submodule of Mλ spanned by all polytabloids et, where t is of shape λ. A polytabloid et is standard if the tableau t is standard. The set of standard polytabloids {et | t is a standard tableau of shape λ} forms a basis of the Specht module Sλ.

On eigenfunctions of the Star Graphs

An embedding M λ into C[Symn]

Let idλ be the standard tableau of shape λ whose rows consist

• f the consecutive elements.

Let Tλ be the set of all tableaux of shape λ. For any tableau t ∈ Tλ, denote by τt the permutation defined be the equation τt(t) = idλ, (3) where τt acts on t by replacing the values of the cells of t. Let us define a linear mapping φ : Mλ → C[Symn]. Since the set

• f all λ-tabloids is a basis for Mλ, it is enough to define images

for λ-tabloids. For any λ-tabloid {t}, where t ∈ Tλ, we put φ({t}) =

• t′∈{t}

τt′.

On eigenfunctions of the Star Graphs

An embedding M λ into C[Symn]

Lemma 1. For any polytabloid et, the equality φ(Jn(et)) = Jn(φ(et)) holds. Lemma 2. Let v ∈ Sλ be an eigenfunction of the operator Jn : Mλ → Mλ with eigenvalue θ. Then φ(v) is an eigenfunction

• f the operator Jn : C[Symn] → C[Symn] with eigenvalue θ.

On eigenfunctions of the Star Graphs

An eigenvector of Jn given by a polytabloid et

Let λ ∈ P(n) be a partition (λ1, λ2, . . . , λs), where s 2, λ1 > λ2 and λi λi+1 for any i ∈ {2, . . . , s − 1}. Put m = λ2 + . . . + λs. In this setting m is the number of cells in all rows of λ but the first. Let t be a standard tableau of shape λ with n placed at its upper right cell. Lemma 3. The polytabloid et is an eigenfunction of Jn with eigenvalue n − m − 1.

On eigenfunctions of the Star Graphs

Expression of the polytabloid eigenfunction

Let et be an eigenfunction from the Lemma 3, and n > 2m holds. For any i ∈ {1, . . . , s} and j ∈ {1, . . . , k}, denote by Rt(i) and Ct(j) the symmetric groups on the elements of ith row and jth column of the tableau t, respectively. Then we have Rt = Rt(1) × Rt(2) × . . . × Rt(s), Ct = Ct(1) × Ct(2) × . . . × Ct(k), where Rt and Ct are the row-stabilizer and the column-stabilizer

• f t, respectively.

On eigenfunctions of the Star Graphs

Expression of the polytabloid eigenfunction

Put CAt = CAt(1) × CAt(2) × . . . × CAt(k), where CAt(j) denotes the subgroup of even permutations in Ct(j). Theorem 3. The equality fφ(et) =

• σ∈Rt(2)×...×Rt(s)
• π∈CAσ(t)

fPπ

(n−m+1,...,n)

holds, where Pπ is a vector of m pairs uniquely determined by π.

On eigenfunctions of the Star Graphs

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On eigenfunctions of the Star Graphs