Closure relations of Newton strata in Iwahori double cosets

We consider the Newton stratification on Iwahori double cosets for a connected reductive group. We prove the existence of Newton strata whose closures cannot be expressed as a union of strata, and show how this is implied by the existence of non-equidimensional affine Deligne–Lusztig varieties. We also give an explicit example for a group of type $$A_4$$ A 4 .

Counting Parabolic Double Cosets in Symmetric Groups

Billey, Konvalinka, Petersen, Solfstra, and Tenner recently presented a method for counting parabolic double cosets in Coxeter groups, and used it to compute $p_n$, the number of parabolic double cosets in $S_n$, for $n\leq13$. In this paper, we derive a new formula for $p_n$ and an efficient polynomial time algorithm for evaluating this formula. We use these results to compute $p_n$ for $n\leq5000$ and to prove an asymptotic formula for $p_n$ that was conjectured by Billey et al.

On growth of double cosets in hyperbolic groups

Let [Formula: see text] be a hyperbolic group, [Formula: see text] and [Formula: see text] be subgroups of [Formula: see text], and [Formula: see text] be the growth function of the double cosets [Formula: see text]. We prove that the behavior of [Formula: see text] splits into two different cases. If [Formula: see text] and [Formula: see text] are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as [Formula: see text]. We can even take [Formula: see text]. In contrast, for quasiconvex subgroups [Formula: see text] and [Formula: see text] of infinite index, [Formula: see text] is exponential. Moreover, there exists a constant [Formula: see text], such that [Formula: see text] for all big enough [Formula: see text], where [Formula: see text] is the growth function of the group [Formula: see text]. So, we have a clear dichotomy between the quasiconvex and non-quasiconvex case.

Parabolic double cosets in Coxeter groups

International audience Parabolic subgroups WI of Coxeter systems (W,S) and their ordinary and double cosets W/WI and WI\W/WJ appear in many contexts in combinatorics and Lie theory, including the geometry and topology of generalized flag varieties and the symmetry groups of regular polytopes. The set of ordinary cosets wWI , for I ⊆ S, forms the Coxeter complex of W , and is well-studied. In this extended abstract, we look at a less studied object: the set of all double cosets WIwWJ for I,J ⊆ S. Each double coset can be presented by many different triples (I, w, J). We describe what we call the lex-minimal presentation and prove that there exists a unique such choice for each double coset. Lex-minimal presentations can be enumerated via a finite automaton depending on the Coxeter graph for (W, S). In particular, we present a formula for the number of parabolic double cosets with a fixed minimal element when W is the symmetric group Sn. In that case, parabolic subgroups are also known as Young subgroups. Our formula is almost always linear time computable in n, and the formula can be generalized to any Coxeter group.

Generic Newton points and the Newton poset in Iwahori-double cosets

We consider the Newton stratification on Iwahori-double cosets in the loop group of a reductive group. We describe a group-theoretic condition on the generic Newton point, called cordiality, under which the Newton poset (that is, the index set for non-empty Newton strata) is saturated and Grothendieck’s conjecture on closures of the Newton strata holds. Finally, we give several large classes of Iwahori-double cosets for which this condition is satisfied by studying certain paths in the associated quantum Bruhat graph.