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The ${\rm SL}_n$-skein algebra of a punctured surface $\mathfrak{S}$, studied by Sikora, is an algebra generated by isotopy classes of $n$-webs living in the thickened surface $\mathfrak{S} \times (-1,1)$, where an $n$-web is a union of framed links and framed oriented $n$-valent graphs satisfying certain conditions. For each ideal triangulation $\lambda$ of $\mathfrak{S}$, L\^e and Yu constructed an algebra homomorphism, called the ${\rm SL}_n$-quantum trace, from the ${\rm SL}_n$-skein algebra of $\mathfrak{S}$ to a so-called balanced subalgebra of the $n$-root version of Fock and Goncharov's quantum torus algebra associated to $\lambda$. We show that the ${\rm SL}_n$-quantum trace maps for different ideal triangulations are related to each other via a balanced $n$-th root version of the quantum coordinate change isomorphism, which extends Fock and Goncharov's isomorphism for quantum cluster varieties. We avoid heavy computations in the proof, by using the splitting homomorphisms of L\^e and Sikora, and a network dual to the $n$-triangulation of $\lambda$ studied by Schrader and Shapiro., Comment: 42 pages

The ${\rm SL}_3$-skein algebra $\mathscr{S}_{\bar{q}}(\mathfrak{S})$ of a punctured oriented surface $\mathfrak{S}$ is a quantum deformation of the coordinate algebra of the ${\rm SL}_3$-character variety of $\mathfrak{S}$. When $\bar{q}$ is a root of unity, we prove the Unicity Theorem for representations of $\mathscr{S}_{\bar{q}}(\mathfrak{S})$, in particular the existence and uniqueness of a generic irreducible representation. Furthermore, we show that the center of $\mathscr{S}_{\bar{q}}(\frak{S})$ is generated by the peripheral skeins around punctures and the central elements contained in the image of the Frobenius homomorphism for $\mathscr{S}_{\bar{q}}(\frak{S})$, a surface generalization of Frobenius homomorphisms of quantum groups related to ${\rm SL}_3$. We compute the rank of $\mathscr{S}_{\bar{q}}(\mathfrak{S})$ over its center, hence the dimension of the generic irreducible representation., Comment: v3: 69 pages, streamlined, added some details and a new section 9

In 1980's H Verlinde suggested to construct and use a quantization of Teichm\"uller spaces to construct spaces of conformal blocks for the Liouville conformal field theory. This suggestion led to a mathematical formulation by Fock in 1990's and later by Fock, Goncharov and Shen, called the modular functor conjecture, based on the Chekhov-Fock quantum Teichm\"uller theory. In 2000's Teschner combined the Chekhov-Fock version and the Kashaev version of quantum Teichm\"uller theory to construct a solution to a modified form of the conjecture. We embark on a direct approach to the conjecture based on the Chekhov-Fock(-Goncharov) theory. We construct quantized trace-of-monodromy along simple loops via Bonahon and Wong's quantum trace maps developed in 2010's, and investigate algebraic structures of them, which will eventually lead to construction and properties of quantized geodesic length operators. We show that a special recursion relation used by Teschner is satisfied by the quantized trace-of-monodromy, and that the quantized trace-of-monodromy for disjoint loops commute in a certain strong sense., Comment: 66 pages; v2: minor updates

For a punctured surface $\mathfrak{S}$, the author and Scarinci (arXiv:2112.13329) have recently constructed a quantization of a moduli space of Lorentzian metrics on the 3-manifold $\mathfrak{S} \times \mathbb{R}$ of constant sectional curvature $\Lambda \in \{-1,0,1\}$. The invariance of this quantization under the action of the mapping class group ${\rm MCG}(\mathfrak{S})$ of $\mathfrak{S}$ yields families of unitary representations of ${\rm MCG}(\mathfrak{S})$ on a Hilbert space, with key ingredients being three versions of the quantum dilogarithm functions depending on $\Lambda$. In this survey article, we review and elaborate on this result., Comment: 32 pages; to appear in the proceedings of the conference in honor of Igor Frenkel's 70th birthday / v2: minor updates

A new canonical Hopf algebra called the quantum pseudo-K\"ahler plane is introduced. This quantum group can be viewed as a deformation quantization of the complex two-dimensional plane $\mathbb{C}^2$ with a pseudo-K\"ahler metric, or as a complexified version of the well-known quantum plane Hopf algebra. A natural class of nicely-behaved representations of the quantum pseudo-K\"ahler plane algebra is defined and studied, in the spirit of the previous joint work of the author and I. B. Frenkel. The tensor square of a unique irreducible representation decomposes into the direct integral of the irreducibles, and the unitary decomposition map is expressed by a special function called the modular double compact quantum dilogarithm, used in the recent joint work of the author and C. Scarinci on the quantization of 3d gravity for positive cosmological constant case. Then, from the associativity of the tensor cube, and from the maps between the left and the right duals, we construct unitary operators forming a new representation of Kashaev's group of transformations of dotted ideal triangulations of punctured surfaces, as an analog of Kashaev's quantum Teichm\"uller theory. The present work thus inspires one to look for a Kashaev-type quantization of 3d gravity for positive cosmological constant., Comment: 36 pages. v2 reflects changes made in the reviewing process. To appear in Commun. Math. Phys

We present a three-dimensional geometric construction of the Virasoro-Bott group, which is a central extension of the group of diffeomorphisms of the circle. Our approach is analogous to the well-known construction of a central extension of the loop group by means of the Wess-Zumino topological term. In particular, the Virasoro-Bott group is realized as a quotient group of diffeomorphisms of the disc with special boundary conditions. We identify the Lie algebra corresponding to our group with the Virasoro algebra. We also show that for generalized boundary conditions the Virasoro algebra is extended to a semidirect product with the Heisenberg algebra. We discuss the relation between our construction, the Chern-Simons theory, and the three-dimensional gravity., Comment: 41 pages. v1: The present paper is a preliminary version, and was mostly written by 2014 / v2: minor changes

Fock-Goncharov's moduli spaces $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ of framed ${\rm PGL}_3$-local systems on punctured surfaces $\frak{S}$ provide prominent examples of cluster $\mathscr{X}$-varieties and higher Teichm\"uller spaces. In a previous paper of the author (arXiv:2011.14765), building on the works of others, the so-called ${\rm SL}_3$ quantum trace map is constructed for each triangulable punctured surface $\frak{S}$ and an ideal triangulation $\Delta$ of $\frak{S}$, as a homomorphism from the stated ${\rm SL}_3$-skein algebra of the surface to a quantum torus algebra that deforms the ring of Laurent polynomials in the cube-roots of the cluster coordinate variables for the cluster $\mathscr{X}$-chart for $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$ associated to $\Delta$. We develop quantum mutation maps between special subalgebras of the cube-root quantum torus algebras for different triangulations and show that the ${\rm SL}_3$ quantum trace maps are natural, in the sense that they are compatible under these quantum mutation maps. As an application, the quantum ${\rm SL}_3$-${\rm PGL}_3$ duality map constructed in the previous paper is shown to be independent of the choice of an ideal triangulation., Comment: v2: title change. reflected revision updates for published version. to appear in Quantum Topol

In this paper we partially settle Fock-Goncharov's duality conjecture for cluster varieties associated to their moduli spaces of ${\rm G}$-local systems on a punctured surface $\frak{S}$ with boundary data, when ${\rm G}$ is a group of type $A_2$, namely ${\rm SL}_3$ and ${\rm PGL}_3$. Based on Kuperberg's ${\rm SL}_3$-webs, we introduce the notion of ${\rm SL}_3$-laminations on $\frak{S}$ defined as certain ${\rm SL}_3$-webs with integer weights. We introduce coordinate systems for ${\rm SL}_3$-laminations, and show that ${\rm SL}_3$-laminations satisfying a congruence property are geometric realizations of the tropical integer points of the cluster $\mathscr{A}$-moduli space $\mathscr{A}_{{\rm SL}_3,\frak{S}}$. Per each such ${\rm SL}_3$-lamination, we construct a regular function on the cluster $\mathscr{X}$-moduli space $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$. We show that these functions form a basis of the ring of all regular functions. For a proof, we develop ${\rm SL}_3$ quantum and classical trace maps for any triangulated bordered surface with marked points, and state-sum formulas for them. We construct quantum versions of the basic regular functions on $\mathscr{X}_{{\rm PGL}_3,\frak{S}}$. The bases constructed in this paper are built from non-elliptic webs, hence could be viewed as higher `bangles' bases, and the corresponding `bracelets' versions can also be considered as direct analogs of Fock-Goncharov's and Allegretti-Kim's bases for the ${\rm SL}_2$-${\rm PGL}_2$ case., Comment: ver2: A major change is that the quantization is added. Some terminology changed, hence the title changed / ver3: Relatively minor corrections, and bibliography update / ver4: Some errors corrected, gaps filled, subsection 5.7 added, conjectural section 6 cut down, referee's feedbacks applied. To appear in Memoirs of the AMS

Quantization of the Teichm\"uller space of a non-compact Riemann surface has emerged in 1980's as an approach to three dimensional quantum gravity. For any choice of an ideal triangulation of the surface, Thurston's shear coordinate functions on the edges form a coordinate system for the Teichm\"uller space, and they should be replaced by suitable self-adjoint operators on a Hilbert space. Upon a change of triangulations, one must construct a unitary operator between the Hilbert spaces intertwining the quantum coordinate operators and satisfying the composition identities up to multiplicative phase constants. In the well-known construction by Chekhov, Fock and Goncharov, the quantum coordinate operators form a family of reducible representations, and the phase constants are all trivial. In the present paper, we employ the harmonic-analytic theory of the Shale-Weil intertwiners for the Schr\"odinger representations, as well as Faddeev-Kashaev's quantum dilogarithm function, to construct a family of irreducible representations of the quantum shear coordinate functions and the corresponding intertwiners for the changes of triangulations. The phase constants are explicitly computed and described by the Maslov indices of the Lagrangian subspaces of a symplectic vector space, and by the pentagon relation of the flips of triangulations. The present work may generalize to the cluster $\mathscr{X}$-varieties.

Quantization of the Teichm\"uller space of a punctured Riemann surface $S$ is an approach to $3$-dimensional quantum gravity, and is a prototypical example of quantization of cluster varieties. Any simple loop $\gamma$ in $S$ gives rise to a natural trace-of-monodromy function $\mathbb{I}(\gamma)$ on the Teichm\"uller space. For any ideal triangulation $\Delta$ of $S$, this function $\mathbb{I}(\gamma)$ is a Laurent polynomial in the square-roots of the exponentiated shear coordinates for the arcs of $\Delta$. An important problem was to construct a quantization of this function $\mathbb{I}(\gamma)$, namely to replace it by a noncommutative Laurent polynomial in the quantum variables. This problem, which is closely related to the framed protected spin characters in physics, has been solved by Allegretti and Kim using Bonahon and Wong's ${\rm SL}_2$ quantum trace for skein algebras, and by Gabella using Gaiotto, Moore and Neitzke's Seiberg-Witten curves, spectral networks, and writhe of links. We show that these two solutions to the quantization problem coincide. We enhance Gabella's solution and show that it is a twist of the Bonahon-Wong quantum trace., Comment: 45 pages. ver2: Author added. Sections 4, 5, statement and proof of main theorem substantially improved / ver3: Changes made for published version have been reflected

In 2006, Fock and Goncharov constructed a nice basis of the ring of regular functions on the moduli space of framed ${\rm PGL}_2$-local systems on a punctured surface $S$. The moduli space is birational to a cluster $\mathcal{X}$-variety, whose positive real points recover the enhanced Teichm\"uller space of $S$. Their basis is enumerated by integral laminations on $S$, which are collections of closed curves in $S$ with integer weights. Around ten years later, a quantized version of this basis, still enumerated by integral laminations, was constructed by Allegretti and Kim. For each choice of an ideal triangulation of $S$, each quantum basis element is a Laurent polynomial in the exponential of quantum shear coordinates for edges of the triangulation, with coefficients being Laurent polynomials in $q$ with integer coefficients. We show that these coefficients are Laurent polynomials in $q$ with positive integer coefficients. Our result was expected in a positivity conjecture for framed protected spin characters in physics and provides a rigorous proof of it, and may also lead to other positivity results, as well as categorification. A key step in our proof is to solve a purely topological and combinatorial ordering problem about an ideal triangulation and a closed curve on $S$. For this problem we introduce a certain graph on $S$, which is interesting in its own right., Comment: 40 pages, 18 figures ; ver2 reflects changes made for journal publication ; to appear in Commun. Math. Phys

Representation theory of the quantum torus Hopf algebra, when the parameter $q$ is a root of unity, is studied. We investigate a decomposition map of the tensor product of two irreducibles into the direct sum of irreducibles, realized as a `multiplicity module' tensored with an irreducible representation. The isomorphism between the two possible decompositions of the triple tensor product yields a map ${\bf T}$ between the multiplicity modules, called the 6j-symbols. We study the left and right dual representations, and correspondingly, the left and right representations on the ${\rm Hom}$ spaces of linear maps between representations. Using the isomorphisms of irreducibles to left and right duals, we construct a map ${\bf A}$ on a multiplicity module, encoding the permutation of the roles of the irreducible representations in the identification of the multiplicity module as the space of intertwiners between representations. We show that ${\bf T}$ and ${\bf A}$ satisfy certain consistency relations, forming a Kashaev-type quantization of the Teichm\"uller spaces of bordered Riemann surfaces. All constructions and proofs in the present work uses only plain representation theoretic language with the help of the notions of the left and the right dual and Hom representations, and therefore can be applied easily to other Hopf algebras for future works., Comment: 37 pages; ver2 reflects changes made for journal publication (to appear in JPAA)

An important problem in the theory of cluster algebras is to compute the fundamental group of the exchange graph. A non-trivial closed loop in the exchange graph, for example, generates a non-trivial identity for the classical and quantum dilogarithm functions. An interesting conjecture, partly motivated by dilogarithm functions, is that this fundamental group is generated by closed loops of mutations involving only two of the cluster variables. We present examples and counterexamples for this naive conjecture, and then formulate a better version of the conjecture for acyclic seeds., Comment: 18 pages, 4 figures; v2: journal version, an appendix added

A cluster variety of Fock and Goncharov is a scheme constructed by gluing split algebraic tori, called seed tori, via birational gluing maps called mutations. In quantum theory, the ring of functions on seed tori are deformed to non-commutative rings, represented as operators on Hilbert spaces. Mutations are quantized to unitary maps between the Hilbert spaces intertwining the representations. These unitary intertwiners are described using the quantum dilogarithm function $\Phi^\hbar$. Algebraic relations among classical mutations are satisfied by the intertwiners up to complex constants. The present paper shows that these constants are $1$. So the mapping class group representations resulting from the Chekhov-Fock-Goncharov quantum Teichm\"uller theory are genuine, not projective. During the course, the hexagon and the octagon operator identities for $\Phi^\hbar$ are derived., Comment: 38 pages; shortened version of arXiv:1602.00361 prepared for journal submission; this ver3 is substantially revised and improved, with the previously conjectural hexagon and the octagon identities proved

A cluster variety of Fock and Goncharov is a scheme constructed from the data related to the cluster algebras of Fomin and Zelevinsky. A seed is a combinatorial data which can be encoded as an $n\times n$ matrix with integer entries, or as a quiver in special cases, together with $n$ formal variables. A mutation is a certain rule for transforming a seed into another seed; the new variables are related to the previous variables by some rational expressions. To each seed one attaches an $n$-dimensional torus, and by gluing the tori along the birational maps defined by the mutation formulas, one constructs a cluster variety. Quantization of a cluster variety assigns to each seed a non-commutative ring which deforms the classical ring of functions on the torus attached to the seed, as well as to each mutation an isomorphism of skew fields of fractions of these non-commutative rings. A representation realizes the non-commutative rings as algebras of operators on Hilbert spaces, and the quantum mutation isomorphisms as unitary maps between the Hilbert spaces that intertwine the operators for the rings. These unitary intertwiners are one of the major results of the Fock-Goncharov quantization of cluster varieties, and are given by the special function called the quantum dilogarithm. The classical mutations satisfy certain algebraic relations, which were known to be satisfied also by the corresponding intertwiners up to complex constants of modulus $1$. The present paper shows by computation that these constants are all $1$. One implication is that the mapping class group representations resulting from the application of the Fock-Goncharov quantization to the quantum Teichm\"uller theory are genuine, not projective., Comment: 56 pages; long version (a shortened version for journal submission is arXiv:1602.00797)

We construct exceptional collections of line bundles of maximal length 4 on $S=(C \times D)/G$ which is a surface isogenous to a higher product with $p_g=q=0$ where $G=G(32,27)$ is a finite group of order 32 having number 27 in the list of Magma library. From these exceptional collections, we obtain new examples of quasiphantom categories as their orthogonal complements., Comment: 18 pages; v2 reflects the revision made during the journal publication process

We define a canonical map from a certain space of laminations on a punctured surface into the quantized algebra of functions on a cluster variety. We show that this map satisfies a number of special properties conjectured by Fock and Goncharov. Our construction is based on the "quantum trace" map introduced by Bonahon and Wong., Comment: 40 pages. Version 2: Removed unjustified assertions about positivity and corrected proofs. Version 3: Listed author affiliations upon request of the second author's institute

We define new coordinates for Fock-Goncharov's higher Teichm\"uller spaces for a surface with holes, which are the moduli spaces of representations of the fundamental group into a reductive Lie group $G$. Some additional data on the boundary leads to two closely related moduli spaces, the $\mathscr{X}$-space and the $\mathscr{A}$-space, forming a cluster ensemble. Fock and Goncharov gave nice descriptions of the coordinates of these spaces in the cases of $G = PGL_m$ and $G=SL_m$, together with Poisson structures. We consider new coordinates for higher Teichm\"uller spaces given as ratios of the coordinates of the $\mathscr{A}$-space for $G=SL_m$, which are generalizations of Kashaev's ratio coordinates in the case $m=2$. Using Kashaev's quantization for $m=2$, we suggest a quantization of the system of these new ratio coordinates, which may lead to a new family of projective representations of mapping class groups. These ratio coordinates depend on the choice of an ideal triangulation decorated with a distinguished corner at each triangle, and the key point of the quantization is to guarantee certain consistency under a change of such choices. We prove this consistency for $m=3$, and for completeness we also give a full proof of the presentation of Kashaev's groupoid of decorated ideal triangulations., Comment: 42 pages, 6 figures

Quantization of universal Teichm\"uller space provides projective representations of the Ptolemy-Thompson group, which is isomorphic to the Thompson group $T$. This yields certain central extensions of $T$ by $\mathbb{Z}$, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension $\hat{T}^{Kash}$ of $T$ resulting from the Kashaev quantization, and show that it corresponds to $6$ times the Euler class in $H^2(T;\mathbb{Z})$. Meanwhile, the braided Ptolemy-Thompson groups $T^*$, $T^\sharp$ of Funar-Kapoudjian are extensions of $T$ by the infinite braid group $B_\infty$, and by abelianizing the kernel $B_\infty$ one constructs central extensions $T^*_{ab}$, $T^\sharp_{ab}$ of $T$ by $\mathbb{Z}$, which are of topological nature. We show $\hat{T}^{Kash}\cong T^\sharp_{ab}$. Our result is analogous to that of Funar and Sergiescu, who computed a presentation of another dilogarithmic central extension $\hat{T}^{CF}$ of $T$ resulting from the Chekhov-Fock(-Goncharov) quantization and thus showed that it corresponds to $12$ times the Euler class and that $\hat{T}^{CF} \cong T^*_{ab}$. In addition, we suggest a natural relationship between the two quantizations in the level of projective representations., Comment: 43 pages, 15 figures. v2: substantially revised from the first version, and the author affiliation changed. // v3: Groups M and T are shown to be anti-isomorphic (new Prop.2.32), which makes the whole construction more natural. And some minor changes // v4: reflects all changes made for journal publication (to appear in Adv. Math.)

We derive the quantum Teichm\"uller space, previously constructed by Kashaev and by Fock and Chekhov, from tensor products of a single canonical representation of the modular double of the quantum plane. We show that the quantum dilogarithm function appears naturally in the decomposition of the tensor square, the quantum mutation operator arises from the tensor cube, the pentagon identity from the tensor fourth power of the canonical representation, and an operator of order three from isomorphisms between canonical representation and its left and right duals. We also show that the quantum universal Teichm\"uller space is realized in the infinite tensor power of the canonical representation naturally indexed by rational numbers including the infinity. This suggests a relation to the same index set in the classification of projective modules over the quantum torus, the unitary counterpart of the quantum plane, and points to a new quantization of the universal Teichm\"uller space., Comment: 41 pages, 9 figures

In this article we give an self contained existence proof for J. Conway's sporadic simple group Co_1 [4] using the second author's algorithm [14] constructing finite simple groups from irreducible subgroups of GL_n(2). Here n = 11 and the irreducible subgroup is the Mathieu group M_{24}. From the split extension E of M_{24} by a uniquely determined 11-dimensional GF(2)M_{24}-module V we construct the centralizer H = C_G(z) of a 2-central involution z of E in an unknown target group G. Then we prove that all the conditions of Algorithm 2.5 of [14] are satisfied. This allows us to construct a simple subgroup G of GL_{276}(23) which we prove to be isomorphic with Conway's original sporadic simple group Co_1 by means of a constructed faithful permutation representation of G and Soicher's presentation [16] of the original Conway group Co_1., Comment: 220 pages

In this article we construct an irreducible simple subgroup G = of GL_{8671}(13) from an irreducible subgroup T of GL_{11}(2) isomorphic to Mathieu's simple group M_{24} by means of Algorithm 2.5 of [13]. We also use the first author's similar construction of Fischer's sporadic simple group G_1 = Fi_{23} described in [11]. He starts from an irreducible subgroup T_1 of GL_{11}(2) contained in T which is isomorphic to M_{23}. In [7] J. Hall and L. S. Soicher published a nice presentation of Fischer's original 3-transposition group Fi_{24} [6]. It is used here to show that G is isomorphic to the simple commutator subgroup Fi'_{24} of Fi_{24}. We also determine a faithful permutation representation of G of degree 306936 with stabilizer G_1 = $\cong$ Fi_{23}. It enabled MAGMA to calculate the character table of G automatically. Furthermore, we prove that G has two conjugacy classes of involutions z and u such that C_G(u) = $\cong$ 2Aut(\Fi_{22}). Moreover, we determine a presentation of H = C_G(z) and a faithful permutation representation of degree 258048 for which we document a stabilizer., Comment: 112 pages

In this article we give self-contained existence proofs for the sporadic simple groups Co_2 and Fi_{22} using the second author's algorithm [10] constructing finite simple groups from irreducible subgroups of GL_n(2). These two sporadic groups were originally discovered by J. Conway [4] and B. Fischer [7], respectively, by means of completely different and unrelated methods. In this article n=10 and the irreducible subgroups are the Mathieu group M_{22} and its automorphism group Aut(M_{22}). We construct their five non-isomorphic extensions E_i by the two 10-dimensional non-isomorphic simple modules of M_{22} and by the two 10-dimensional simple modules of A_{22} = Aut(M_{22}) over F=GF(2). In two cases we construct the centralizer H_i = C_{G_i}(z_i) of a 2-central involution z_i of E_i in any target simple group G_i. Then we prove that all the conditions of Algorithm 7.4.8 of [11] are satisfied. This allows us to construct G_3 $\cong$ Co_2 inside GL_{23}(13) and G_2 $\cong$ Fi_{22} inside GL_{78}(13). We also calculate their character tables and presentations., Comment: 94 pages

In the first section of this senior thesis the author provides some new efficient algorithms for calculating with finite permutation groups. They cannot be found in the computer algebra system MAGMA, but they can be implemented there. For any finite group G with a given set of generators, the algorithms calculate generators of a fixed subgroup of G as short words in terms of original generators. Another new algorithm provides such a short word for a given element of G. In the later sections, the author gives a self-contained existence proof for Fischer's sporadic simple group Fi_{23} using G. Michler's Algorithm [11] constructing finite simple groups from irreducible subgroups of GL_n(2). This sporadic group was originally discovered by B. Fischer in [6] by investigating 3-transposition groups, see also [5]. This thesis gives a representation theoretic and algorithmic existence proof for his group. The author constructs the three non-isomorphic extenstions E_i by the two 11-dimensional non-isomorphic simple modules of the Mathieu group M_{23} over F=GF(2). In two cases Michler's Algorithm fails. In the third case the author constructs the centralizer H=C_G(z) of a 2-central involution z of E_i in any target simple group G. This allows the author to construct G inside GL_{782}(17). Its four generating matrices, character table and representatives for conjugacy classes are computed. It follows that G and Fi_{23} have the same character table., Comment: 102 pages; undergraduate senior thesis in Cornell (May 2008)