Research

Work-in-progress

My field of research stretches between combinatorics and representation theory of finite classical groups, which I would like to widen by considering their interactions with probability. In my Ph.D. thesis, I studied the centers of symplectic group ring algebras along the lines of Farahat-Higman and solved an open question posed by Wang et. al. in an AIM paper.  I have formulated the corresponding theorem on my own and solved it independently. Prof. Wang, Univ. of Virginia, kindly participated in my committee as the external advisor on my Ph.D. committee as the subject matter expert who provided the scientific evaluation of my Ph.D. research.  Recently I submitted a paper on the double coset algebras of the wreath products of symmetric groups. As part of my research, I will study the asymptotic behaviors of Hecke algebras of finite groups of Lie type following Farahat and Higman’s work in addition to Ivanov and Kerov’s work.

1- As my Ph.D thesis I investigated the structure constants of the center H(n) of the integral group algebra of the symplectic group Sp(2n,q). The reflection length on the group GL(2n,q) induces a filtration on the algebras H(n). I proved that the structure constants of the associated filtered algebra are independent of n, which is known as “stability of the family” in the literature following Wan and Wang (2019). As a technical tool in the proof, I determined the growth of the intersection of the centralizers of two elements with respect to the identification of Sp(n,q) in Sp(n+r,q). To this end, I introduced a new concept “primitive symplectic centralizer”, which allowed me to black-box quadratic equations and reduce certain counting questions into linear ones. Arxiv version of this study can be found here: https://arxiv.org/abs/1812.04720. (Now published at J. Algebra)
2- Upon finishing my dissertation I studied the double coset algebras H(n) of certain wreath products in the symmetric groups. In his book “Symmetric functions and Hall polynomials”, I.G. MacDonald proves that a certain bi-colored graph attached to permutations can be used to parameterize the double cosets of the hyperoctahedral group. In my study, I generalized the bi-colored graph defined in ibid to parameterize the double cosets of the wreath products in the symmetric group and also introduced “minimal representatives of double cosets”. Using minimal representatives, I showed that structure coefficients of the algebras H(n) are polynomials in n. I also proved that the sizes of the connected components of the graphs that parameterize the double-cosets induce a filtration in the double coset algebra. In the course of this proof, I introduced a new concept that I denominated “discrete graph evolution”. I then used this filtration to prove a similar stability result. Arxiv version of this study can be found here: https://arxiv.org/abs/1910.10480. (Review and resubmit at Journal of Algebraic Combinatorics)

RESEARCH PROJECTS:  Below is the list of projects I plan to carry out. I introduce first the projects that are at a higher stage of progress and then those relatively less established.

1- (This project is now close to an end.) Using Farahat-Higman approach I plan to investigate the double coset algebra H(n) of the symplectic group Sp(2n,q) in the general linear group GL(2n,q). The expected output of this project will be to prove a stability result for the family H(n). Proving such a theorem requires to establish the following:

a- Parameterization of the Sp(2n,q)-double cosets in GL(2n,q). Such work is necessary to parameterize the double coset algebra of the limiting groups, namely the double cosets of the “infinite symplectic group” Sp(q) in the “infinite general linear group” GL(q), which are both defined as the direct limit of finite-dimensional counterparts of these groups. One parametrization of the double cosets is given by Goldstein and Guralnick in their paper entitled “Alternating forms and self-adjoint operators”. More specifically, they prove that the conjugacy classes of GL(n,q) parameterize the double cosets of Sp(2n,q) in GL(2n,q). The specific elements that Goldstein and Guralnick used to parameterize the double-cosets can serve as the variant of the minimal coset representative, which I considered in my past work indicated at the second item above. Hence, I expect to obtain

a polynomiality result with relatively less effort. Still, this requires a detailed analysis of the intersection of the centralizer of a linear map with the symplectic group. In particular, one needs to obtain a result like Corollary 4.18 of my latest study. Proving such a theorem involves determining the growth of certain subgroups of the symplectic group as n grows. I have tested several cases and observed that the concept of the primitive symplectic centralizer of an isometry, which I discussed earlier, can be generalized to general linear maps and can be used to determine the aforementioned growth.

b- Find an appropriate filtration that would yield a filtered algebra structure on the double-coset algebra.  The preliminary calculations I have done suggest that the reflection length of the conjugacy class in GL(2n,q) representing a given Sp(2n,q)-double coset may serve the desired filtration. This anticipation is also supported by a similar result concerning the corresponding Weyl groups. It is known that the partitions of n can be used to parameterize the B_n-double cosets. In their paper entitled “Generators of the Hecke algebra of (S_2n, B_n)”, Aker and Can proved that the transposition length of the conjugacy class in S_n that represents a given B_n-double coset induces a filtration in the attached double coset algebra.

c- Relate the Sp(n,q) centralizers of general linear maps x and xy when the ‘weight’ of the double cosets of x and y adds up to that of xy. Recall that if the sum of the reflection lengths of x and y is equal to the reflection length of xy then the fixed space of xy is equal to the intersection of the fixed spaces of x and y. One needs to generalize this theorem in the setting of the Sp(2n,q)-double cosets. A similar result in the setting of the double cosets of the wreath products is already proved by myself, cf. Prop.7.18 of “Stability of the Hecke algebra of wreath products”.  I believe that those arguments, which are ‘coordinate specific’, can be generalized to the symplectic-double coset case.

Timeline: I intend to finalize this first project in 2020.

2- Using Ivanov-Kerov’s approach I plan to investigate the Hecke algebras which I considered in my past research. In the Farahat-Higman approach, one needs to prove the so-called stability theorem in order to obtain a universal algebra H (called the Farahat-Higman ring of the family) that projects onto the algebras H(n) under consideration. However, in this approach, there are no ‘connecting morphisms’ between the algebras H(n) that would realize H as a projective limit. One of the aims of the Ivanov-Kerov approach is to overcome this anomaly and hence replace H(n) with a suitable algebra P(n) satisfying the following properties:

i- The algebras H(n) and P(n) have the same representation-theoretic significance for the groups under consideration;
ii- The algebras P(n) form an inverse system and the corresponding inverse limit P is isomorphic to the Farahat-Higman ring H.

Construction of the algebras P(n) requires finding a suitable semi-group that captures the representation theory of the underlying group, which at the same time captures the ‘missing’ information that prevents the existence of connecting morphisms in the group algebras considered in the Farahat-Higman approach. The property (ii) listed above suggests that a stability result is an indication of the existence of a semi-group that satisfy the properties listed above. In fact, a stability result has been proven for all those cases where the Ivanov-Kerov approach has been successfully implemented. The semi-group in the case of symmetric groups is given by the partial permutation, a concept introduced by Ivanov and Kerov, while the Farahat-Higman approach was established directly by Farahat and Higman. In the case of general linear groups over finite fields, the corresponding ‘semi-group’ of partial isomorphisms over finite fields is constructed by P.L. Méliot (2014), while the stability result, in this case, is proved by J. Wan and W. Wang (2019). Finally, the work of Aker and Can (2012), and the work of Can and myself (2017) established the stability result for the double-coset algebra of the hyperoctahedral group, while the work of O. Tout (2018) establishes the same result by finding the correct semi-group. However, the semi-groups in the cases that I proved the stability theorems are yet to be constructed.

Thus, my second project consists of two sub-projects. Firstly, I plan to generalize the work of Méliot to the case of the center of the integral group rings over symplectic groups. Secondly, I plan to generalize the work of Tout to the double cosets of wreath products. To this end, I need to work over and learn the details of the Ivanov-Kerov method in each case.
Timeline: While I plan to master my knowledge on the Ivanov-Kerov method in 2020/21, I intend to finalize this second project during 2021. The approach of Ivanov-Kerov also admits probabilistic applications to random characters. I aim to broaden my scope by learning these probabilistic applications and find out the parallel applications in the cases I plan to investigate.

The remaining projects are at a preliminary stage and I plan to develop the ideas during my post-doc:

3- Joint with Atabey Kaygun (https://web.itu.edu.tr/kaygun/). We plan to investigate the possibilities of implementing the Farahat-Higman approach to the quantum permutation groups.

4- Joint with Ayhan Günaydin (http://web.boun.edu.tr/ayhan.gunaydin/). In this project, we seek to understand the ways of extending the asymptotic studies listed above to the general linear group GL(n,F), where F is a local field of characteristic zero.