arXiv subjects

Tathagata Basak

Publications and source records attributed to Tathagata Basak.

17 recordsLinked to original sources

arXiv2026

Fundamental racks of braid spaces of complex reflection groups

Let $\Gamma$ be a complex reflection group acting on the complex affine or hyperbolic space $X$ with the set of reflecting hyperplanes $\mathcal{H}$. We define an augmented rack $(G, \mathcal{K}, p)$ associated to the orbifold fundamental group $G := \pi_1^{\operatorname{orb}}( \Gamma \backslash (X - \mathcal{H}))$ which plays the role of the fundamental rack of a framed link complement as defined by Fenn and Rourke. This yields representations of the orbifold fundamental group $G$ on the cohomology of the associated rack space.

math.GT↗

arXiv2026

Petersen graph and monodromy of the 27 lines on the Clebsch surface

Let $G$ be the orbifold fundamental group of the moduli space of smooth cubic surfaces $\mathcal{M}_{\mathsf{sm}}$ in $\mathbb{P}^3_{\mathbb{C}}$ with base point at the Clebsch surface $X_{\mathbf{1}}$. The image of the monodromy action $G \to \lbrace \text{Permutations of $27$ lines on $X_{\mathbf{1}}$} \rbrace$ is famously the Weyl group of type $E_6$. Here we give a description of this monodromy action in terms of the Petersen graph by working out the action of ten explicit generators of $G$ by elementary calculation. These ten generators were found in joint work with Allcock and Looijenga while studying the description of $\mathcal{M}_{\mathsf{sm}}$ as a discriminant complement in a complex $4$-ball quotient.

math.AG↗

arXiv2023

The Deligne-Mostow 9-ball, and the monster

The "monstrous proposal" of the first author is that the quotient of a certain 13-dimensional complex hyperbolic braid group, by the relations that its natural generators have order 2, is the bimonster" (M x M)semidirect Z/2. Here M is the monster simple group. We prove that this quotient is either the bimonster or Z/2. In the process, we give new information about the isomorphism found by Deligne-Mostow, between the moduli space of 12-tuples in CP1 and a quotient of the complex 9-ball. Namely, we identify which loops in the 9-ball quotient correspond to the standard braid generators.

math.GT↗

arXiv2022

Exact factorizations and extensions of finite tensor categories

We extend \cite{G} to the nonsemisimple case. We define and study exact factorizations $\B=\A\bullet \C$ of a finite tensor category $\B$ into a product of two tensor subcategories $\A,\C\subset \B$, and relate exact factorizations of finite tensor categories to exact sequences of finite tensor categories with respect to exact module categories \cite{EG}. We apply our results to study exact factorizations of quasi-Hopf algebras, and extensions of a finite group scheme theoretical tensor category \cite{G2} by another one. We also provide several examples to illustrate our results.

math.QA↗

arXiv2018

A new complex reflection group in $PU(9,1)$ and the Barnes-Wall lattice

We show that the projectivized complex reflection group $Γ$ of the unique $(1+i)$-modular Hermitian $\mathbb{Z}[i]$-module of signature $(9,1)$ is a new arithmetic reflection group in $PU(9,1)$. We find $32$ complex reflections of order four generating $Γ$. The mirrors of these $32$ reflections form the vertices of a sort of Coxeter-Dynkin diagram $D$ for $Γ$ that encode Coxeter-type generators and relations for $Γ$. The vertices of $D$ can be indexed by sixteen points and sixteen affine hyperplanes in $\mathbb{F}_2^4$. The edges of $D$ are determined by the finite geometry of these points and hyperplanes. The group of automorphisms of the diagram $D$ is $2^4 \colon (2^3 \colon L_3(2)) \colon 2$. This group transitively permutes the $32$ mirrors of generating reflections and fixes an unique point $τ$ in $\mathbb{C} H^9$. These $32$ mirrors are precisely the mirrors closest to $τ$. These results are strikingly similar to the results satisfied by the complex hyperbolic reflection group at the center of Allcock's monstrous proposal.

math.RT↗

arXiv2017

Root space decomposition of $\mathfrak{g}_2$ from octonions

We describe a simple way to write down explicit derivations of octonions that form a Chevalley basis of $\mathfrak{g}_2$. This uses the description of octonions as a twisted group algebra of the finite field $\mathbb{F}_8$. Generators of $\operatorname{Gal}(\mathbb{F}_8/\mathbb{F}_2)$ act on the roots as $120$-degree rotations and complex conjugation acts as negation.

math.RT↗

arXiv2017

A Poincaré series on hyperbolic space

Let $L$ be the unique even self-dual lattice of signature $(25,1)$. The automorphism group $\operatorname{Aut}(L)$ acts on the hyperbolic space $\mathcal{H}^{25}$. We study a Poincaré series $E(z,s)$ defined for $z$ in $\mathcal{H}^{25}$, convergent for $\operatorname{Re}(s) > 25$, invariant under $\operatorname{Aut}(L)$ and having singularities along the mirrors of the reflection group of $L$. We compute the Fourier expansion of $E(z,s)$ at a "Leech cusp" and prove that it can be meromorphically continued to $\operatorname{Re}(s) > 25/2$. Analytic continuation of Kloosterman sum zeta functions imply that the individual Fourier coefficients of $E(z,s)$ have meromorphic continuation to the whole $s$-plane.

math.RT↗

arXiv2017

The octonions as a twisted group algebra

We show that the octonions can be defined as the $\mathbb{R}$-algebra with basis $\lbrace e^x \colon x \in \mathbb{F}_8 \rbrace$ and multiplication given by $e^x e^y = (-1)^{φ(x,y)}e^{x + y}$, where $φ(x,y) = \operatorname{tr}(y x^6)$. While it is well known that the octonions can be described as a twisted group algebra, our purpose is to point out that this is a useful description. We show how the basic properties of the octonions follow easily from our definition. We give a uniform description of the sixteen orders of integral octonions containing the Gravesian integers, and a computation-free proof of their existence.

math.RA↗

arXiv2017

Generators for a complex hyperbolic braid group

We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic $13$-space, take the quotient of the remaining space by a discrete group, and find generators for the orbifold fundamental group of the quotient. These generators have the most natural form: loops corresponding to the hyperplanes which come nearest the basepoint. Our results support the conjecture that motivated this study, the "monstrous proposal", which posits a relationship between this braid group and the monster finite simple group.

math.GT↗

arXiv2015

Indicators of Tambara-Yamagami categories and Gauss sums

We prove that the higher Frobenius-Schur indicators, introduced by Ng and Schauenburg, give a strong enough invariant to distinguish between any two Tambara-Yamagami fusion categories. Our proofs are based on computation of the higher indicators as quadratic Gauss sums for certain quadratic forms on finite abelian groups and relies on the classification of quadratic forms on finite abelian groups, due to Wall. As a corollary to our work, we show that the state-sum invariants of a Tambara-Yamagami category determine the category as long as we restrict to Tambara-Yamagami categories coming from groups G whose order is not a power of 2. Turaev and Vainerman proved this result under the assumption that G has odd order and they conjectured that a similar result should hold for groups of even order. We also give an example to show that the assumption that G does not have a power of 2, cannot be completely relaxed.

math.QA↗

arXiv2015

Geometric generators for braid-like groups

We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from n dimensional complex vector space, or n dimensional complex hyperbolic space, or the Hermitian symmetric space for O(2,n), and then takes the quotient by a discrete group $PΓ$. The classical example is the braid group, but there are many similar "braid-like" groups that arise in topology and algebraic geometry. Our main result is that if $PΓ$ contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certain property, then G is generated by the analogues of the generators of the classical braid group. We apply this to obtain generators for G in a particular intricate example in complex hyperbolic space of dimension 13. The interest in this example comes from a conjectured relationship between this braid-like group and the monster simple group M, that gives geometric meaning to the generators and relations in the Conway-Simons presentation of $(M \times M):2$.

math.GT↗

arXiv2012

Modular lattices from finite projective planes

Using the geometry of the projective plane over the finite field F_q, we construct a Hermitian Lorentzian lattice L_q of dimension (q^2 + q + 2) defined over a certain number ring $\cO$ that depends on q. We show that infinitely many of these lattices are p-modular, that is, p L'_q = L_q, where p is some prime in $\cO$ such that |p|^2 = q. The reflection group of the Lorentzian lattice obtained for q = 3 seems to be closely related to the monster simple group via the presentation of the bimonster as a quotient of the Coxeter group on the incidence graph of P^2(F_3). The Lorentzian lattices L_q sometimes lead to construction of interesting positive definite lattices. In particular, if q is a rational prime that is 3 mod 4, and (q^2 + q + 1) is norm of some element in Q[\sqrt{-q}], then we find a 2q(q+1) dimensional even unimodular positive definite integer lattice M_q such that Aut(M_q) contains PGL(3,F_q). We find that M_3 is the Leech lattice.

math.RT↗

arXiv2012

The complex Lorentzian Leech lattice and the bimonster (II)

Let $D$ be the incidence graph of the projective plane over $\FF_3$. The Artin group of the graph $D$ maps onto the bimonster and a complex hyperbolic reflection group $Γ$ acting on 13 dimensional complex hyperbolic space $Y$. The generators of the Artin group are mapped to elements of order 2 (resp. 3) in the bimonster (resp. $Γ$). Let $Y^{\circ} \subseteq Y$ be the complement of the union of the mirrors of $Γ$. Daniel Allcock has conjectured that the orbifold fundamental group of $Y^{\circ}/Γ$ surjects onto bimonster. In this article we study the reflection group $Γ$. Our main result shows that there is homomorphism from the Artin group of $D$ to the orbifold fundamental group of $Y^{\circ}/Γ$, obtained by sending the Artin generators to the generators of monodromy around the mirrors of the generating reflections in $Γ$. This answers a question in Allcock's article "A monstrous proposal" and takes a step towards the proof of Allcock's conjecture. The finite group $\op{PGL}(3, \FF_3) \subseteq \Aut(D)$ acts on $Y$ and fixes a complex hyperbolic line pointwise. We show that the restriction of $Γ$-invariant meromorphic automorphic forms on $Y$ to the complex hyperbolic line fixed by $\op{PGL}(3, \FF_3)$ gives meromorphic modular forms of level 13.

math.GR↗

arXiv2010

On Coxeter Diagrams of complex reflection groups

We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $\cE = \ZZ[e^{2 πi/3}]$: there are only four such lattices, namely, the $\cE$-lattices whose real forms are $A_2$, $D_4$, $E_6$ and $E_8$. Next, we address the issue of characterizing the diagrams for unitary reflection groups, a question that was raised by Broué, Malle and Rouquier. To this end, we describe an algorithm which, given a unitary reflection group $G$, picks out a set of complex reflections. The algorithm is based on an analogy with Weyl groups. If $G$ is a Weyl group, the algorithm immediately yields a set of simple roots. Experimentally we observe that if $G$ is primitive and $G$ has a set of roots whose $\ZZ$--span is a discrete subset of the ambient vector space, then the algorithm selects a minimal generating set for $G$. The group $G$ has a presentation on these generators such that if we forget that the generators have finite order then we get a (Coxeter-like) presentation of the corresponding braid group. For some groups, such as $G_{33}$ and $G_{34}$, new diagrams are obtained. For $G_{34}$, our new diagram extends to an "affine diagram" with $\ZZ/7\ZZ$ symmetry.

math.GR↗

arXiv2008

Combinatorial cell complexes and Poincare duality

We define and study a class of finite topological spaces, which model the cell structure of a space obtained by gluing finitely many Euclidean convex polyhedral cells along congruent faces. We call these finite topological spaces, combinatorial cell complexes (or c.c.c). We define orientability, homology and cohomology of c.c.c's and develop enough algebraic topology in this setting to prove the Poincare duality theorem for a c.c.c satisfying suitable regularity conditions. The definitions and proofs are completely finitary and combinatorial in nature.

math.AT↗

arXiv2006

Reflection group of the quaternionic Lorentzian Leech lattice

In this article we study a second example of the phenomenon studied in "Complex Lorentzian Leech lattice and bimonster".(Arxiv. math.GR/0508228). The results and methods of proof are similar. We find 14 roots in the automorphism group of the the quaternionic Lorentzian Leech lattice L that form the Coxeter diagram D given by the incidence graph of projective plane over finite field of order 2. We prove that the reflections in these 14 roots generate the automorphism group of L. We find evidence that these reflections behave like the simple roots and the vector fixed by the diagram automorphisms behaves like the Weyl vector for the reflection group. Much of the work follows the analogy that D is like the Coxeter-Dynkin diagram for this hyperbolic reflection group.

math.GR↗

arXiv2006

The complex Lorentzian Leech lattice and the bimonster

We find 26 reflections in the automorphism group of the the Lorentzian Leech lattice L over Z[exp(2*pi*i/3)] that form the Coxeter diagram seen in the presentation of the bimonster. We prove that these 26 reflections generate the automorphism group of L. We find evidence that these reflections behave like the simple roots and the vector fixed by the diagram automorphisms behaves like the Weyl vector for the refletion group.

math.GR↗