Philip Puente. Statement of Research and Scholarly Interests 1 Areas of Interest

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1 Philip Puente Statement of Research and Scholarly Interests 1 Areas of Interest I work on questions on reflection groups at the intersection of algebra, geometry, and topology. My current research uses techniques from the theory of hyperplane arrangements, lattices in complex space, and invariant theory to study fundamental groups. 2.1 Braid Conjecture 2 Dissertation In my dissertation at the University of North Texas, I address a well known result proven for all finite reflection groups and all affine Weyl groups in the case of complex reflection group of infinite order. Theorem 1. Let M W be the space of regular orbits for the action of a reflection group W. Then a presentation for the fundamental group π 1 (M W /W ) is obtained from the presentation of W by removing the finite order relations of the generators. Many people have worked on this result: Brieskorn [5] and Artin [1], on finite Coxeter groups, Bannai [2], Broué, Rouquier, Malle [6], and Bessis [4], on finite complex reflection groups, Viet Dung using a CW semi-cell complex [10], on infinite Weyl groups (affine Weyl groups), Jon McCammond s student Benjamin Coté (Ph.D. Thesis 2016), on the exceptional infinite complex reflection group corresponding to G 4, Myself (Ph.D. thesis 2017), on almost all infinite complex reflection groups. The work in my thesis extends Viet Dung s semi-cell construction to one over C using Voronoi regions. This approach is different than Broué, Malle, Rouquier, Bessis (and Brieskorn, Artin, and others) and provides a uniform method. I combine this approach with known results of Broué, Malle, Rouquier, Bessis to obtain a presentation of the fundamental group for most of the complex reflection groups of infinite order. The fundamental group of the space of regular orbits of a reflection group W is called the braid group associated to W. In general, it is not an easy task to compute these fundamental groups, but it was noticed that the braid group associated to a finite real reflection group W has a presentation that can be obtained from the Coxeter presentation of W one need only remove the condition that generators have finite order. Noticing Theorem 1, this observation was also proven to hold for all the infinite real reflection groups and all the finite complex reflection groups. One wonders if the same is true for infinite complex reflection groups when we do not assume the group is a complexification of a real group (genuine). In 1998, Malle conjectured the following: Conjecture 1. (Malle [11]) A presentation of the braid group of a genuine infinite complex reflection group W can be obtained by removing the finite order relation of the generators of the Coxeter-like presentation of W. 1

2 In [12], Popov classified the infinite complex reflection groups. There is one infinite family consisting of the groups [G(r, p, n)] k which are symmetries of the regular r-polytope in C n extended by a lattice and the rest are a finite collection of exceptional groups. Malle verified his conjecture for some set of infinite groups. The main results of my thesis are concerned with the infinite families. Theorem 2. Let W be an infinite complex reflection group. Then the presentation for the braid group B W is obtained from the presentation of W by removing the finite order relations of the generators if Lin(W ) = G(r, p, n) excepting 7 cases. The five excluded cases above are the groups [G(4, 2, 2)] 3, [G(6, 2, 2)] 2, [G(6, 3, 2)] 2, [G(4, 2, n)], [G(3, 3, n)], [G(4, 4, n)], and [G(6, 6, n)]. The first three groups are 2 dimensional and likely satisfy the conjecture while the last four are infinite families. I am checking these now as a modified argument is required in calculation of their braid groups. 2.2 Reflection Groups and Proof Techniques Now let V = R n or V = C n with n < and let A(V ) be the set of all invertible affine transformations of V. A reflection is an affine transformation of finite order with fixed point space of codimension 1. A reflection group is a discrete group that is generated by reflections. Given a reflection group W, we denote by Lin(W ) the subgroup of GL(V ) generated by all the linear parts of the elements of W and by Tran(W ) the subgroup of W that consists of purely translational elements, i.e., v v + b for some b V. Reflection groups contain many familiar groups such as Weyl groups and Coxeter groups such as the infamous E 8 group. The finite irreducible real reflection groups were classified by Coxeter, and the infinite real reflection groups are in fact the affine Weyl groups. In 1954, Shephard and Todd [13] classified all the finite complex reflection groups into one infinite family of groups parameterized by three numbers r, p, n and 34 exceptional groups. Each member of the infinite family is denoted by G(r, p, n) and consists of the n by n monomial matrices with nonzero entries that are r th roots of unity such that the product of all of these non zero entries is an (r/p) th root of unity. In 1982 [12], Popov classified the infinite complex reflection groups. He showed that infinite complex reflection groups do not behave as nicely as do infinite real reflection groups. For all infinite real groups, it is always true that W = Lin(W ) Tran(W ), but there are some infinite complex reflection groups that are not such a semidirect product. Now let W be a reflection group acting on the vector space V and let H be the union of hyperplanes that are the fixed point spaces of reflections in W and let M W = V H. We define the braid group associated to the group W to be the group B W = π 1 (M W /W ), that is, the braid group is the fundamental group of the orbit space of M W under W. In [10], Viet Dung used chambers and alcoves of real reflection groups to carry out his calculations of the braid group of infinite real reflection groups. This is where the first difficulty arose. Complex reflection groups do not have chambers or alcoves. Therefore, I created a fundamental domain for every complex reflection group that mimics the role served of a chamber or alcove of a given real reflection group by the use of Voronoi regions. Another difficulty that arose was the 2

3 following. In order to use his techniques, the natural projection p : M W M W /W must be a regular covering map. To ensure this, every member of M W must have trivial stabalizer in W, i.e., the only element of W that fixes a point of M W must be 1 V. A famous theorem of Steinberg [14] states that given a finite complex reflection group, the stabilizer of a point must a reflection group, i.e., the stabilizer in W of point is generated by reflections that fix the point. This is not true for infinite complex reflection groups, the stabilizer need not be generated by reflections. The following theorem addresses this fact. Theorem 3. Suppose W is a infinite complex reflection group acting on C n such that Lin(W ) = G(r, p, n). The stabilizer in W of v C n not contained in a reflection hyperplane of W is trivial if and only if W is one of the following groups: [G(3, 1, n)] 1, [G(4, 1, n)] 1,2, [G(6, 1, n)], [G(4, 2, n)] 1,2, [G(4, 2, 2)] 3, [G(6, 3, n)] 1, [G(6, 2, n)] 2, and [G(6, 2, 2)] 2. With this observation, the braid group is computed by first calculating the fundamental group of the space of regular orbits. The space of regular orbits consist of the orbits under W of all the points of V that are not fixed by a nontrivial element of W. Therefore, I needed to prove the following fact that shows that the fundamental group of the space of regular orbits is in fact the braid group. Theorem 4. Let W be a reflection group that acts on V with N W the set of all points with trivial stabilizer. Then π 1 (M W /W ) = π 1 (N W /W ). This observation allows one to carry the techniques mentioned above through to compute π(n W /W ) which in the case of infinite complex reflection groups is the braid group π(m W /W ) and prove the main result of my thesis Theorem 2. 3 Future Work 3.1 Completely Resolve Braid Conjecture Benjamin Coté and I plan to address the case of 2 dimensional complex reflection groups by combining techniques he used in [8], and with the use of computational software, it will be possible to completely resolve Malle s conjecture for all infinite complex reflection groups, i.e., for the finite number of exceptional cases remaining. It will also be possible to extend Theorem 1 and include all these exceptional cases. 3.2 Complete Search for Regular Points for Infinite Reflection Groups Coté determined that there is an exceptional infinite complex reflection group for which the space of regular points is not the complement of the set of hyperplanes [8]. Thus, I will attempt to completely resolve the statement given in Theorem 3 when W is an arbitrary infinite complex reflection group. 3.3 Zariski Van Kampen Theorem for Analytic Varieties Bessis used a variant of the Zariski Van Kampen theorem to compute the braid group of a finite complex reflection group [3]. It would be interesting to investigate this theorem for complex analytic varieties. 3

4 3.4 Polynomial Invariants One of the most beautiful results in the theory of finite reflection groups is the Chevalley- Shephard-Todd Theorem: the space of invariant polynomials of a finite group G is a polynomial algebra if and only if G is a reflection group. That is, the space of invariant polynomials of a finite group G is generated by polynomials that are algebraically independent if and only if G is a reflection group. Now every infinite complex reflection group has elements that are translations, i.e., v v +b. Therefore, the space of invariant polynomials must be trivial, but there is a way to translate the result above about finite reflection groups to the setting of infinite reflection groups. Let G be a finite group acting on C n. Then it also acts on P n 1 (C), the projective space of C n. The Chevallay-Shephard-Todd theorem above can be stated as the following. Theorem 5. P n 1 (C)/G is isomorphic to a weighted projective space iff G is a reflection group. In this new formulation, there is an analog for infinite complex reflection groups. Conjecture 2 ([9] Dolgachev). Let W be an infinite complex reflection group acting on C n. Then the space C n /W is isomorphic to a weighted projective space. It would be interesting to use the techniques of complex analytic varieties and fundamental domains to investigate this theorem. 3.5 Quaternionic Reflection Group Cohen [7] classified all the finite quaternionic reflection group. It would interesting to classify the infinite quaternionic reflection in the spirit of Popov [12] and see if we can formulate similar results, i.e., invariant lattices, braid groups, etc. 3.6 Intuitive Description of Braid Groups It would be interesting to give an intuitive description of the braid groups in the spirit of Artin [1], that is, in terms of braiding diagrams or some other combinatorial device. Is it possible to describe all braid groups this way? References [1] E. Artin. Theory of braids. Ann. of Math. (2), 48: , [2] E. Bannai. Fundamental groups of the spaces of regular orbits of the finite unitary reflection groups of dimension 2. J. Math. Soc. Japan, 28(3): , [3] D. Bessis. Variations on Van Kampen s method. J. Math. Sci. (N. Y.), 128(4): , Geometry. [4] D. Bessis and J. Michel. Explicit presentations for exceptional braid groups. Experiment. Math., 13(3): , [5] E. Brieskorn. Die Fundamentalgruppe des Raumes der regulären Orbits einer endlichen komplexen Spiegelungsgruppe. Invent. Math., 12:57 61, [6] M. Broué, G. Malle, and R. Rouquier. Complex reflection groups, braid groups, Hecke algebras. J. Reine Angew. Math., 500: , [7] A. M. Cohen. Finite quaternionic reflection groups. J. Algebra, 64(2): ,

5 [8] B. Coté. A complex Euclidean reflection group and its braid group. PhD thesis, Univeristy of California, Santa Barbara. [9] I. V. Dolgachev. Reflection groups in algebraic geometry. Bull. Amer. Math. Soc. (N.S.), 45(1):1 60, [10] N. V. Dung. The fundamental groups of the spaces of regular orbits of the affine Weyl groups. Topology, 22(4): , [11] G. Malle. Presentations for crystallographic complex reflection groups. Transform. Groups, 1(3): , [12] V. L. Popov. Discrete complex reflection groups, volume 15 of Communications of the Mathematical Institute, Rijksuniversiteit Utrecht. Rijksuniversiteit Utrecht, Mathematical Institute, Utrecht, [13] G. C. Shephard and J. A. Todd. Finite unitary reflection groups. Canadian J. Math., 6: , [14] R. Steinberg. Differential equations invariant under finite reflection groups. Trans. Amer. Math. Soc., 112: ,