arXiv subjects

Sean Rostami

Publications and source records attributed to Sean Rostami.

6 recordsLinked to original sources

arXiv2017

Conjugation of Transitive Permutation Pairs and Dessins d'Enfants

Let E be a finite set. Given permutations x and y of E that together generate a transitive subgroup, for which s is it true that x and the conjugate of y by s also generate a transitive subgroup? Such transitive permutation pairs encode dessins d'enfants, important graph-theoretic objects which are also known to have great arithmetic significance. The absolute Galois group acts on dessins d'enfants and permutes them in a very mysterious way. Two dessins d'enfants that share certain elementary combinatorial features are related by conjugations as above, and dessins d'enfants in the same Galois-orbit share these features and more, so it seems worthwhile to have a good answer to the above question. I classify, relative to x and y, exactly those transpositions s for which the new pair is guaranteed to be transitive. I also provide examples of the "exceptional" s which show the range of possible behavior and prove that the above question for the exceptional cases is equivalent to a natural question about deletion in graphs that may have a good answer in this more structured world of topological graphs. Finally, I classify transpositions s according to how they change the genus of the surface underlying the dessin d'enfant of x, y. Some of the tools, like the Reroute Operation/Theorem, may have use beyond Dessins d'Enfants.

math.CO↗

arXiv2016

Kottwitz's nearby cycles conjecture for a class of unitary Shimura varieties

This paper proves that the nearby cycles complexes on a certain family of PEL local models are central with respect to the convolution product of sheaves on the corresponding affine flag varieties. As a corollary, the semisimple trace functions defined using the action of Frobenius on those nearby cycles complexes are, via the sheaf-function dictionary, in the centers of the corresponding Iwahori-Hecke algebras. This is commonly referred to as Kottwitz's Conjecture. The reductive groups associated to the PEL local models under consideration are unramified unitary similitude groups with even dimension. The proof follows the method of Haines-Ngo 2002. Upon completion of the first version of this paper, Pappas and Zhu released a preprint, now published, which contained within its scope the main theorem of this paper. However, the methods of Pappas-Zhu are very different and some of the proofs from this paper have been useful in forthcoming work of Haines-Stroh.

math.AG↗

arXiv2016

On the Canonical Representatives of a Finite Weyl Group

Let K be a field and G a split connected reductive affine algebraic K-group. Let T be a split maximal torus of G, W its finite Weyl group, and R its root system. After fixing a realization of R in G and choosing a simple system for R, one gets a system of representatives for W in G(K), called the Canonical Representatives. It is well-known that these representatives rarely form a subgroup, and it is necessary for some questions to understand and quantify this failure. Various new formulas are given which constitute progress in this direction. An application of such formulas to the simple supercuspidals of Gross-Reeder and Reeder-Yu is provided.

math.RT↗

arXiv2014

Conjugacy classes of non-translations in affine Weyl groups and applications to Hecke algebras

Let W be an Iwahori-Weyl group of a connected reductive group G over a non-archimedean local field. I prove that if w is an element of W that does not act on the corresponding apartment of G by a translation then one can apply to w a sequence of conjugations by simple reflections, each of which is length-preserving, resulting in an element w' for which there exists a simple reflection s such that l(sw's)>l(w'). Even for affine Weyl groups, a special case of Iwahori-Weyl groups and also an important subclass of Coxeter groups, this is a new fact about conjugacy classes. Further, there are implications for Iwahori-Hecke algebras H of G: one can use this fact to give dimension bounds on the "length-filtration" of the center Z(H), which can in turn be used to prove that suitable linearly-independent subsets of Z(H) are a basis.

math.RT↗

arXiv2014

The Bernstein presentation for general connected reductive groups

Let F be a non-Archimedean local field and let G be a connected reductive affine algebraic F-group. Let I be an Iwahori subgroup of G(F) and denote by H(G; I) the Iwahori-Hecke algebra, i.e. the convolution algebra of complex-valued functions on G(F) which are left- and right-invariant by I-translations. This article proves that the Iwahori-Hecke algebra H(G; I) has both an Iwahori-Matsumoto Presentation and a Bernstein Presentation analogous to those for affine Hecke algebras on root data found in Lusztig's "Affine Hecke algebras and their graded version", and gives a basis (in other words, an explicit Bernstein Isomorphism) for the center Z[H(G; I)] also analogous to that found in loc. cit.

math.RT↗

arXiv2009

The Satake isomorphism for special maximal parahoric Hecke algebras

Let G denote a connected reductive group over a nonarchimedean local field F. Let K denote a special maximal parahoric subgroup of G(F). We establish a Satake isomorphism for the Hecke algebra H of K-bi-invariant compactly supported functions on G(F). The key ingredient is a Cartan decomposition describing the double coset space K\G(F)/K. We also describe how our results relate to the treatment of Cartier, where K is replaced by a special maximal compact open subgroup K' of G(F) and where a Satake isomorphism is established for the Hecke algebra of K'-bi-invariant compactly supported functions on G(F).

math.RT↗