Yuval Roichman

Random Cayley graphs and expanders

For every 1 > δ > 0 there exists a c = c(δ) > 0 such that for every group G of order n, and for a set S of c(δ) log n random elements in the group, the expected value of the second largest eigenvalue of the normalized adjacency matrix of the Cayley graph X(G, S) is at most (1 - δ). This implies that almost every such a graph is an ϵ(δ)-expander. For Abelian groups this is essentially tight, and explicit constructions can be given in some cases.

The Flag Major Index and Group Actions on Polynomial Rings

A new extension of the major index, defined in terms of Coxeter elements, is introduced. For the classical Weyl groups of type B, it is equidistributed with length. For more general wreath products it appears in an explicit formula for the Hilbert series of the (diagonal action) invariant algebra.

Descent representations and multivariate statistics

Combinatorial identities on Weyl groups of types A and B are derived from special bases of the corresponding coinvariant algebras. Using the Garsia-Stanton descent basis of the coinvariant algebra of type A we give a new construction of the Solomon descent representations. An extension of the descent basis to type B, using new multivariate statistics on the group, yields a refinement of the descent representations. These constructions are then applied to refine well-known decomposition rules of the coinvariant algebra and to generalize various identities.

Equi-distribution over descent classes of the hyperoctahedral group

A classical result of MacMahon shows that the length function and the major index are equidistributed over the symmetric group. Foata and Schutzenberger gave a remarkable refinement and proved that these parameters are equi-distributed over inverse descent classes, implying bivariate equi-distribution identities. Type B analogues of these results, refinements and consequences are given in this paper.

Expansion Properties of Cayley Graphs of the Alternating Groups

LetCbe a conjugacy class in the alternating groupAn, and let supp(C) be the number of nonfixed digits under the action of a permutation inC. For every 1>?>0 andn?5 there exists a constantc=c(?)>0 such that if supp(C)??nthen the undirected Cayley graphX(An, C) is acexpander. A family of such Cayley graphs withsupp(C)=o(n)is not a family ofc-expanders. For every?>0, ifsupp(C)?nthen sets of vertices of order at most(12??)(n?(n/supp(C)))!inX(An, C) expand. The proof of the last result combines spectral and representation theory techniques with direct combinatorial arguments.

Signed Mahonians

A classical result of MacMahon gives a simple product formula for the generating function of major index over the symmetric group. A similar factorial-type product formula for the generating function of major index together with sign was given by Gessel and Simion. Several extensions are given in this paper, including a recurrence formula, a specialization at roots of unity and type B analogues.

Alternating subgroups of Coxeter groups

We study combinatorial properties of the alternating subgroup of a Coxeter group, using a presentation of it due to Bourbaki.

Statistics on wreath products and generalized binomial-stirling numbers

Various statistics on wreath products are defined via canonical words, “colored” right to left minima and “colored” descents. It is shown that refined counts with respect to these statistics have nice recurrence formulas of binomial-Stirling type. These extended Stirling numbers determine (via matrix inversion) dual systems, which are also shown to have combinatorial realizations within the wreath product. The above setting also gives rise to a MacMahon-type equi-distribution theorem over subsets with prescribed statistics.

Diameter of graphs of reduced words and galleries

For finite reflection groups of types A and B, we determine the diameter of the graph whose vertices are reduced words for the longest element and whose edges are braid relations. This is deduced from a more general theorem that applies to supersolvable hyperplane arrangements.

On the Achromatic Number of Hypercubes

The achromatic number of a finite graph G, ?(G), is the maximum number of independent sets into which the vertex set may be partitioned, so that between any two parts there is at least one edge. For an m-dimensional hypercube Pm2 we prove that there exist constants 0