Sunil K. Chebolu Illinois State University Joint work ...people.cst.cmich.edu/salis1bt/algcomb/2014fall/slides_sunil.pdf · Sunil K. Chebolu Illinois State University Joint work withKeir

Group Algebras and Circulant Bipartite Graphs

Sunil K. CheboluIllinois State University

Joint work with Keir Lockridge and Gail YamskulnaarXiv:1404.4096

Algebra and Combinatorics Seminar: September 17, 2014

1/1 Sunil K.Chebolu Group Algebras and Circulant Bipartite Graphs

Overview

I A student in my elementary number theory class asked aquestion about Zn which seemed to be very naive.

I The question led to some questions about the structure ofunits in a group algebra.

I Our theorems which address these questions involve Mersenneand 2-rooted primes.

I We were able to translate our theorems into statements aboutcirculant bipartite graphs.

I This gave us combinatorial characterizations of Mersenneprimes and 2-rooted primes!!

Continuation of this work led to other fundamental questionsabout fields (yesterday’s talk) and Fuchs’ problems.


3 parts of this talk:

1. The genesis of this research.

2. Group algebras : Mersenne primes and 2-rooted primes.

3. Circulant bipartite graphs

This research is inspired by two number theory courses I taught atIllinois State University.

I MAT 330 – Spring 2011

I MAT 410 – Fall 2013


The genesis of this research

Number Theory (MAT 330): Introduced Zn - the set ofcongruence classes mod n.

Write down the multiplication tables for Z3 and Z4.

Here is the multiplication table for Z8.

Z8 :

∗ 0 1 2 3 4 5 6 7

0 0 0 0 0 0 0 0 01 0 1 2 3 4 5 6 72 0 2 4 6 0 2 4 63 0 3 6 1 4 7 2 54 0 4 0 4 0 4 0 45 0 5 2 7 4 1 6 36 0 6 4 2 0 6 4 27 0 7 6 5 4 3 2 1


Elliott asked: I see that 1’s in these multiplication tables appearonly on the diagonal. Is that always true?

No! Consider Z5.(2)(3) = 1 in Z5.

For what values of n do 1’s occur only on the diagonal in themultiplication table of Zn, never off the diagonal?

The word “diagonal” refers to “the main diagonal.”


Theorem (C. 2012) The multiplication table for Zn contains 1’sonly on the diagonal if and only if n is a divisor of 24.

I gave 5 proofs of this theorem:

I The Chinese remainder theorem

I Dirichlet’s theorem on primes in an arithmetic progression

I The structure theory of units in Zn

I Bertrand-Chebyshev theorem

I Erdos-Ramanujan theorem


What about other rings?

A ring R is a ∆2 ring if every unit u in R is such that u2 = 1.

Theorem (C., Mayers, 2013) The polynomial ring Zn[x ] is a∆2-ring if and only if n divides 12.

Let p be a prime. Say that a ring R is a ∆p ring if every unit u inR is such that up = 1.

This definition does not give any interesting results for Zn or Zn[x ].


Group Algebras

Let us look at group algebras.

I k – field

I G – group

I kG – ring of all formal linear combination of the elements ofG with coefficients from k .

Theorem (CLY-14) kG is a ∆2 ring if and only if kG is either F2C r2

or F3C r2 for some 0 < r ≤ ∞.

Theorem (CLY-14) Let G be an abelian group and p be an oddprime. kG is a ∆p-ring if and only if p is a Mersenne prime andkG is either F2(C r

p) or Fp+1(C rp) for some 0 < r ≤ ∞.


An interesting part of the theorem is the following special case:

Theorem (CLY-14) The group algebra F2Cp is a ∆p ring if andonly if p is Mersenne prime.

We shall give a proof of this special case now.

This special case leads to several other characterizations ofMersenne primes involving:

I Binomial coefficients

I Circulant matrices

I Bipartite graphs.


Theorem (CLY-14) Let x be a generator of Cp (p > 3). p is aMersenne prime if and only if (1 + x + x2)p = 1 in F2Cp.

Theorem (CLY-14) Let p > 3 be a prime. p is Mersenne if andonly if

[circ(1, 1, 0, 0, · · · , 0)]p = [circ(1, 0, 0, 1, 0, · · · 0)]p mod 2

Theorem (CLY-14) Let p > 3 be a prime. p is Mersenne if andonly if (

p

r

)≡(

p

3r modp

)mod 2 ∀ 1 ≤ r ≤ p − 1.

Our first main theorem gives 12 characterizations of Mersenneprimes!


Circulant Bipartite Graphs

p odd prime.

A := {a0, a1, a2, · · · , ap−1} and B := {b0, b1, b2, · · · , bp−1}

A (p, p) bipartite graph is a graph where all edges go between Aand B.

Let M = (mij) be the biadjacency matrix of G . (mij = 1 if ai isadjacent to bj and 0 otherwise.)

G is a circulant bipartite graph if M is a circulant matrix.


A (5, 5) circulant bipartite graph:

a0

a1

a2

a3

a4

b0

b1

b2

b3

b4 1 0 0 1 11 1 0 0 11 1 1 0 00 1 1 1 00 0 1 1 1


Perfect Matchings: A perfect matching of G is a collection ofedges of G which set up a 1-1 correspondence between A and B(a.k.a 1-factor).

Example:


A counting problem

p odd prime.

A := {a0, a1, · · · , ap−1} and B := {b0, b1, · · · , bp−1}

How many circulant bipartite graphs on A and B have an oddnumber of perfect matchings?

Call this number Λp.


A closed formula for Λp:

Theorem (CLY-14) The number Λp of (p, p) circulant bipartitegraphs (labeled) with an odd number of perfect matchings is givenby

Λp = (2ordp2 − 1)p−1ordp2 ,

where ordp2 is the smallest positive integer r such that 2r ≡ 1mod p.

Example: p = 7. What is ord72? 21 = 2, 22 = 4, 23 = 1. Thereforeord72 = 3. This means Λ7 = (23 − 1)6/3 = 72 = 49.

We will sketch a proof of this result but first let us see someconsequences.


Corollary (CLY-14)Λp ≤ 2p−1 − 1

This follows immediately from the formula for Λp

Direct proof

I Total number of circulant bipartite graphs on A and B is 2p.( =⇒ Λp ≤ 2p)

I Exactly half of these graphs have an even degree. An evendegree (p, p) circulant bipartite graph cannot have an oddnumber of perfect matchings. (why?) ( =⇒ Λp ≤ 2p−1)

I The complete bipartite graph has p! perfect matchings( =⇒ Λp ≤ 2p−1 − 1).


Graph theoretic characterization of 2-rooted primes

Theorem (CLY-14)Λp = 2p−1 − 1

if and only if 2 is a primitive root mod p (i.e. ordp2 = p − 1)

Question Will the equality in the above corollary hold for infinitelymany primes?

Equivalently, is 2 a primitive root for infinitely many primes?


Artin’s Conjecture: If a is an integer which is not equal to −1 andnot a perfect square, then a will be a primitive root mod p forinfinitely many primes.

This is a very deep and important conjecture in number theory.

There is not even a single value of a for which this conjecture isresolved.

The Riemann Hypothesis =⇒ Artin’s conjecture !

In 1984, R. Gupta and M. Ram Murty showed unconditionally thatArtin’s conjecture is true for infinitely many values of a using sievemethods


Proof of the counting formula

I There is a natural 1-1 correspondence between (p, p) bipartitegraphs and (p × p) binary matrices.

I Circulant (p, p) bipartite graphs correspond to (p × p)circulant binary matrices.

I Circulant (p, p) bipartite graphs with odd number of perfectmatchings correspond to invertible (p × p) circulant binarymatrices! (We shall see why this is true shortly.)

I Key point: The latter has a natural group structure.


Let M be the biadjacency matrix of a (p, p)-bipartite graph G .

Define the permanent of M as

Perm(M) =∑π∈Sp

m1π(1)m2π(2) · · ·mpπ(p)

Every perfect matching in G corresponds to a permutation π in Sp

such that miπ(i) = 1 for all i . Therefore this formula is countingexactly the number of perfect matchings in G .

det(T ) ≡ perm(T ) mod 2.

Lemma G has an odd number of perfect matchings if and only ifthe biadjacency matrix M of G is invertible mod 2.


G := (p, p) circulant bipartite graphs which have an odd number ofperfect matchings

circp(F2)∗ := the set of invertible p × p circulant matrices.

Then we have the following isomorphisms:

G ∼= circp(F2)∗ ∼= (F2Cp)∗ ∼=(

F2[x ]

(xp − 1)

)∗Recall that F2[x]

(xp−1) is a product of finite fields. It decomposes as a

product of p − 1/ordp2 copies of the finite field F2ordp2 .

|G| = (2ordp2 − 1)p−1ordp2 (= Λp).

Note: Now G is equipped with a natural group structure!


Moral of this theorem:

When working with a collection of mathematical objects oneshould always see if the collection in question is associated withsome natural algebraic structure.


Constructing circulant bipartite graphs with odd number of perfectmatchings.

Proposition (CLY-14) Let f (x) be an irreducible polynomial overthe field of 2 elements such that 1 < deg f (x) 6= ord(2, p). Thenthe (p, p) circulant bipartite graph in which a0 is adjacent to bj foreach j such that x j is a non-zero term in f (x) will have an oddnumber of perfect matchings.

Example The irreducible polynomial 1 + x + x2 in F2[x ] gives

a0

a1

a2

a3

a4

b0

b1

b2

b3

b4


Graph theoretic characterization of Mersenne primes

Theorem (CLY-14) p is Mersenne if and only if every circulant(p, p) bipartite graph with odd number of perfect matchings has

sij(p) mod 2 = δij for all 1 ≤ i , j ≤ n,

where sij(p) is the number of pseudopaths from ai to bj of lengthp.

Note: This pseudopath condition is obtained (and equivalent to)from

Ap = Ip mod 2


A pseudopath of length 3 from a0 to b1.


A new and a simple proof of an well-known result in number theory.

Theorem 3 is the only prime number which is both Mersenne and2-rooted.

Standard proof in the literature uses the Quadratic ReciprocityLaw.

Proof A double counting argument

U := solutions of the equation xp = 1 in F2Cp.

p Mersenne =⇒ |U| = (2ordp2 − 1)p−1ordp2 − 1 = 2p−1 − 1

p 2-rooted =⇒ |U| = p

2p−1 − 1 = p ⇐⇒ p = 3


Thank you!

Questions?


Documents

Sunil K. Chebolu Illinois State University Joint work ...people.cst.cmich.edu/salis1bt/algcomb/2014fall/slides_sunil.pdf · Sunil K. Chebolu Illinois State University Joint work withKeir