Character theory of symmetric groups, subgroup growth of Fuchsian groups, and random walks

We develop a number of statistical aspects of symmetric groups (mostly dealing with the distribution of cycles in various subsets of Sn), asymptotic properties of (ordinary) characters of symmetric groups, and estimates for the multiplicities of root number functions of these groups. As main applications, we present an estimate for the subgroup growth of an arbitrary Fuchsian group, a finiteness result for the number of Fuchsian presentations of such a group (resolving a long-standing problem of Roger Lyndon), as well as a proof of a well-known conjecture of Roichman concerning the mixing time of random walks on symmetric groups.     

Multiplicity-free representations of symmetric groups

Building on work of Saxl, we classify the multiplicity-free permutation characters of all symmetric groups of degree 66 or more. A corollary is a complete list of the irreducible characters of symmetric groups (again of degree 66 or more) which may appear in a multiplicity-free permutation representation. The multiplicity-free characters in a related family of monomial characters are also classified. We end by investigating a consequence of these results for Specht filtrations of permutation modules defined over fields of prime characteristic.     

A generalization of Coxeter groups,root systems,and Matsumoto’s theorem

The root systems appearing in the theory of Lie superalgebras and Nichols algebras admit a large symmetry extending properly the one coming from the Weyl group. Based on this observation we set up a general framework in which the symmetry object is a groupoid. We prove that in our context the groupoid is generated by simple reflections and Coxeter relations. In a broad sense this answers a question of Serganova. Our weak version of the exchange condition allows us to prove Matsumoto’s theorem. Therefore the word problem is solved for the groupoid.     

Generalized blocks for symmetric groups

We study, via character-theoretic methods, an ℓ-analogue of the modular representation theory of the symmetric group, for an arbitrary integer ℓ≥2. We find that many of the invariants of the usual block theory (ie. when ℓ is prime) generalize in a natural fashion to this new context. Oblatum 21-III-2002 & 5-VIII-2002?Published online: 8 November 2002     

Large dimension homomorphism spaces between Specht modules for symmetric groups

Let F be a field of characteristic p. We show that Hom_FΣn(S^λ,S^μ) can have arbitrarily large dimension as n and p grow, where S^λ and S^μ are Specht modules for the symmetric group Σ_n. Similar results hold for the Weyl modules of the general linear group. Every previously computed example has been at most one-dimensional, with the exception of Specht modules over a field of characteristic two. The proof uses the work of Chuang and Tan, providing detailed information about the radical series of Weyl modules in Rouquier blocks.     

Arithmetic of characters of generalized symmetric groups

The result here answers the following questions in the affirmative: Can the Galois action on all abelian (Galois) fields $K/\mathbb{Q}$ be realized explicitly via an action on characters of some finite group? Are all subfields of a cyclotomic field of the form $\mathbb{Q}(\chi)$, for some irreducible character $\chi$ of a finite group G? In particular, we explicitly determine the Galois action on all irreducible characters of the generalized symmetric groups. We also determine the smallest extension of $\mathbb{Q}$ required to realize (using matrices) a given irreducible representation of a generalized symmetric group. Received: 18 February 2002     

Nichols-Woronowicz model of coinvariant algebra of complex reflection groups

We give a model of the coinvariant algebra of the complex reflection groups as a subalgebra of a braided Hopf algebra called Nichols-Woronowicz algebra.     

Euler homology

We geometrically construct a homology theory that generalizes the Euler characteristic mod 2 to objects in the unoriented cobordism ring of a topological space X. This homology theory Eh _* has coefficients in every nonnegative dimension. There exists a natural transformation that for X = pt assigns to each smooth manifold its Euler characteristic mod 2. The homology theory is constructed using cobordism of stratifolds, which are singular objects defined below. An isomorphism of graded -modules is shown for any CW-complex X. For discrete groups G, we also define an equivariant version of the homology theory Eh _*, generalizing the equivariant Euler characteristic.     

Orbifold genera, product formulas and power operations

We generalize the definition of orbifold elliptic genus and introduce orbifold genera of chromatic level h, using h-tuples rather than pairs of commuting elements. We show that our genera are in fact orbifold invariants, and we prove integrality results for them. If the genus arises from an H_∞-map into the Morava-Lubin-Tate theory E_h, then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold M in terms of the genus of M itself. Our formula is the p-typical analogue of the Dijkgraaf-Moore-Verlinde-Verlinde formula for the orbifold elliptic genus [R. Dijkgraaf et al., Elliptic genera of symmetric products and second quantized strings Comm. Math. Phys. 185(1) (1997) 197-209]. It depends only on h and not on the genus.     

A linear backward Euler scheme for the saturation equation: Regularity results and consistency

We consider a linearization of a numerical scheme for the saturation equation (or porous medium equation) , through first order expansions of the fractional function f and the inverse of the function , after a regularization of the porous medium equation. We establish a regularity result for the Continuous Galerkin Method and a regularity result for the linearized scheme analogous to the corresponding nonlinear scheme. We then show that the linearized scheme is consistent with the nonlinear scheme analyzed in a previous work.     

Pathologies in cohomology of non-paracompact Hausdorff spaces

We construct a non-paracompact Hausdorff space for which ?ech cohomology does not coincide with sheaf cohomology. Moreover, the sheaf of continuous real-valued functions is neither soft nor acyclic, and our space admits non-numerable principal bundles.     

Localization and completion of nilpotent groups of automorphisms

We study nilpotent subgroups of automorphism groups in the category of groups and the homotopy category of spaces. We establish localization and completion theorems for nilpotent groups of automorphisms of nilpotent groups. We then apply these algebraic theorems to prove analogous results for certain groups of self-homotopy equivalences of spaces.     

Braces and symmetric groups with special conditions

We study left braces satisfying special conditions, or identities. We are particularly interested in the impact of conditions like Raut and lri on the properties of the left brace and its associated solution of the Yang–Baxter equation (YBE). We show that the solution $(G,r_G)$ of the YBE associated to the structure group $G=G(X,r)$ (with the natural structure of a left brace) of a nontrivial solution $(X,r)$ of the YBE has multipermutation level 2 if and only if G satisfies lri. It is known that every (left) brace with lri satisfies condition Raut. We prove that for a graded Jacobson radical ring with no elements of additive order two the conditions lri and Raut are equivalent. We construct a finite two-sided brace with condition Raut which does not satisfy lri. We show that a finitely generated two-sided brace which satisfies lri has a finite multipermutation level which is bounded by the number of its generators.     

A note on Cartan matrices for symmetric groups

Using generating functions, a very simple explicit formula for the determinants of the p-Cartan matrices of symmetric groups is given. Our method works also when p is a composite number.Received: 5 September 2001     

The condition number of the Schur complement in domain decomposition

Summary. It is shown that for elliptic boundary value problems of order 2m the condition number of the Schur complement matrix that appears in nonoverlapping domain decomposition methods is of order , where d measures the diameters of the subdomains and h is the mesh size of the triangulation. The result holds for both conforming and nonconforming finite elements. Received: January 15, 1998     

Left symmetric left distributive operations on a group

We study equational theories of several left symmetric left distributive operations on groups. Normal forms of terms in the variety of LSLD groupoids, LSLD medial groupoids, LSLD idempotent groupoids and LSLD medial idempotent groupoids are found. Received October 11, 2001; accepted in final form December 9, 2004.