A GENERATING FUNCTION OF THE NUMBER OF HOMOMORPHISMS FROM A SURFACE GROUP INTO A FINITE GROUP

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1 A GENERATING FUNCTION OF THE NUMBER OF HOMOMORPHISMS FROM A SURFACE GROUP INTO A FINITE GROUP MOTOHICO MULASE AND JOSEPHINE T. YU 2 Abstract. A generating function of the number of homomorphisms from the fundamental group of a compact oriented or non-orientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate the number of homomorphisms using the decomposition of the group algebra into irreducible factors. This gives a new proof of the classical formulas of Frobenius Schur and Mednykh. 0. Introduction Let S be a compact oriented or non-orientable surface without boundary and χ(s) its Euler characteristic. The subect of our study is a generating function of the number Hom(π (S) G) of homomorphisms from the fundamental group of S into a finite group G. We give a generating function in terms of a non-commutative integral Eqn.(2.7) or Eqn.(3.2) according to the orientability of S. The idea of such integrals comes from random matrix theory. Our integrals can be thought of as a generalization of real symmetric complex hermitian and quaternionic self-adoint matrix integrals. The graphical expansion methods for real symmetric [7 4] and complex hermitian [4] matrix integrals are generalized in [3] for quaternionic self-adoint matrix integrals. The technique developed in [3] is further generalized in [32] to the integrals over matrices with values in non-commutative -algebras. In this article we consider matrix integrals over group algebras. Surprisingly the graphical expansion of the integral gives a generating function of Hom(π (S) G) for all closed surfaces.. Counting Formulas Computation of our generating functions Eqn.(2.7) and Eqn.(3.2) yields a new proof of the following classical counting formulas: Theorem. (Mednykh [24]). Let G be a finite group of order and Ĝ the set of all complex irreducible representations of G. By V λ we denote the irreducible representation parameterized by λ Ĝ. Then for every compact Riemann surface S we have (.) (dim V λ ) χ (S) χ (S) Hom(π (S) G) λ Ĝ where χ(s) is the Euler characteristic of the surface S. Remark.2. For a surface of genus 0 Eqn.(.) reduces to the classical formula (dim V λ ) 2. λ Ĝ Date: November This article is based on the first author s talks at CIMAT Guanauato and MSRI Berkeley in September 2002 and serves as an announcement of [32]. Research supported by NSF grant DMS and UC Davis. 2 Research supported by NSF grant VIGRE DMS and UC Davis.

2 2 MOTOHICO MULASE AND JOSEPHINE YU Since dim V λ is a divisor of we note that the expression χ (S) (dim V λ ) χ (S) Hom(π (S) G) / λ Ĝ is an integer for a surface of positive genus. For a non-orientable surface there is another formula: Theorem.3 (Frobenius-Schur [3]). Let us decompose Ĝ into three disoint subsets according to the Frobenius-Schur indicator: Ĝ λ Ĝ } χ λ (γ 2 ) Ĝ 2 λ Ĝ } (.2) χ λ (γ 2 ) 0 Ĝ 3 λ Ĝ } χ λ (γ 2 ). Then we have (.3) (dim C V λ ) χ (S) + ( dim C V λ ) χ (S) χ (S) Hom(π (S) G) λ Ĝ λ Ĝ3 where S is now an arbitrary compact non-orientable surface without boundary. Remark.4. If we take S RP 2 then χ(rp 2 ) and π (RP 2 ) Z/2Z and the formula reduces to a well-known formula [7 37] dim C V λ dim C V λ the number of involutions in G. λ Ĝ λ Ĝ3 In particular if every complex irreducible representation of G is defined over R then Ĝ Ĝ and the same formula (.) holds for an arbitrary closed surface S orientable or nonorientable. This class of groups includes symmetric groups S n. 2. Orientable Case The generating function of Hom(π (S) G) for an oriented surface S comes from analysis of hermitian matrix integrals. The prototype of the asymptotic expansion formula is (2.) log e 2 NtrX2 e N t 3 trx dµ(x) H NC Γ connected Ribbon graph Aut R Γ Nχ (S Γ ) 3 t v (Γ) for an N N hermitian matrix integral [4] where H NC denotes the space of hermitian matrices of size N and e N 2 trx2 dµ(x) the normalized probability measure on H NC R N 2. A ribbon graph is a graph with a cyclic order assigned to each vertex. Equivalently it is a graph Γ that is drawn on a closed oriented surface S such that the complement S \ Γ is the disoint union of open disks. We use f(γ) to denote the number of these disks (or faces) and v (Γ) the number of -valent vertices of Γ. Let v(γ) v (Γ) and e(γ) v (Γ) 2

3 HOMOMORPHISMS FROM A SURFACE GROUP INTO A FINITE GROUP 3 be the number of vertices and edges of Γ respectively. Then the genus g(s) of the surface S is given by a formula for Euler characteristic χ(s) 2 2g(S) v(γ) e(γ) + f(γ). The automorphism group Aut R (Γ) of a ribbon graph Γ consists of automorphisms of the celldecomposition of S that is determined by the graph. We refer to [30] for precise definition of ribbon graphs and their automorphism groups. The asymptotic formula is an equality in the ring Q(N)[[t 3 t 4 t 5... ]] of formal power series in an infinite number of variables t 3 t 4 t 5... with coefficients in the field Q(N) of rational functions in N. The size N of matrices is considered as a variable here. The topology of this formal power series ring is the Krull topology defined by setting deg t. The monomial tv (Γ) is a finite product for each connected ribbon graph Γ and has degree 2e(Γ). The matrix integral in LHS of Eqn.(2.) is meaningful only as an asymptotic series. Using the asymptotic technique developed in [28 29] we can obtain a well-defined formal power series in Q(N)[[t 3 t 4 t 5... ]] from the calculation of the integral. First we consider a truncation of variables (t 3 t 4... t 2m ) for some m. If we replace the integral with Z(m) H NC e 2 NtrX2 e N 2m t 3 trx dµ(x) then it converges and defines a holomorphic function on (t 3... t 2m t 2m ) C 2m 3 t 2m C Re(t 2m ) < 0}. The holomorphic function has a unique asymptotic series expansion at (t 3... t 2m t 2m ) 0. We can show that the terms of degree n in the asymptotic expansion of Z(m) are stable for every m > n. Therefore lim m Z(m) is convergent in the Krull topology of Q(N)[[t 3 t 4 t 5... ]] and determines a well-defined element. LHS of Eqn.(2.) is the logarithm of this limit. Let us now consider a finite-dimensional -algebra A together with a linear map called trace : A C satisfying that ab ba. A typical example is the group algebra A C[G] of a finite group G. We define a -operation by : C[G] x x γ γ x x γ γ C[G]. As a trace we use the character of the regular representation χ reg by linearly extending it to the whole group algebra. Let denote the real vector subspace of C[G] consisting of self-adoint elements. We need a Lebesgue measure on. The self-adoint condition x x means x γ x γ. Let us decompose the group G into the disoint union of three subsets (2.2) G G I G + G where G I γ G γ 2 } is the set of involutions of G and G + and G are chosen so that G \ G I G + G (2.3) (G + ) G.

4 4 MOTOHICO MULASE AND JOSEPHINE YU Every self-adoint element of C[G] is written as (2.4) x x γ γ + ( yγ (γ + γ ) + iz γ (γ γ ) ) 2 I + where x γ y γ and z γ are real numbers. Thus we have R as a real vector space. We see from Eqn.(2.4) that if x is self-adoint then (2.5) χ reg(x 2 ) (x γ ) 2 ( + (y γ ) 2 + (z γ ) 2). I + This is a non-degenerate quadratic form on R. Let dx denote the Lebesgue measure on that is invariant under the Euclidean transformations with respect to the quadratic form (2.5). A normalized Lebesgue measure is defined by (2.6) dµ(x) dx exp ( 2 χ reg(x 2 ) ) dx. Theorem 2.. As a formal power series in infinitely many variables t 3 t 4 t 5... we have an equality ( (2.7) log exp ) 2 χ reg(x 2 ) exp t χ reg(x ) dµ(x) 3 Aut R Γ χ (S Γ ) Hom(π (S Γ ) G) t v (Γ) 3 Γ connected ribbon graph with valence 3 where S Γ is the oriented surface determined by a ribbon graph Γ. Proof. In the computation of the integral it is easier to use a normalized trace function χ reg instead of χ reg. The formula to be established is then ( (2.8) log exp ) 2 x2 exp t x dµ(x) 3 Aut R Γ f(γ) Hom(π (S Γ ) G) t v (Γ) 3 Γ connected ribbon graph with valence 3 where f(γ) is the number of faces in the surface S Γ. First we expand the exponential factor: exp t x ( v t ) v 3 v 3 v 4 v 5...! v x v 3 The coefficient of tv in the asymptotic expansion is then ( (2.9) v! v exp ) 2 x2 x v dµ(x). 3 We note that (2.9) is a convergent integral. Let y yγ γ C[G] be a variable running on the group algebra and write y γ u γ + iw γ as the sum of real and imaginary 3.

5 HOMOMORPHISMS FROM A SURFACE GROUP INTO A FINITE GROUP 5 parts. Now introduce (2.0) y y γ γ where y γ 2 ( ) u γ i w γ Lemma 2.2. For every > 0 and n 0 we have ( ) n (2.) e x(y+y ) x n e x(y+y ). y. Proof. Since the y derivative of y is zero we can ignore y in the formula. ( ) e xy y y γ y γ γ γ e γ xγ y γ γ...γ x γ γ...γ x e xy. x γ Applying this computation n times we obtain (2.). Using the completion of square we have γ γ e xy (2.2) 2 x2 + x(y + y ) 2 (x (y + y )) (y + y ) 2. Thus (2.3) exp ( ) 2 x2 x n dµ(x) 3 ( ) n y ( ) n y ( ) n e y e e 2 (y+y ) 2 2 x2 + x(y+y ) dµ(x) y0 2 (x (y+y )) (y+y ) 2 dµ(x) y0 y0 where we used the fact that y + y and the translational invariance of the Lebesgue measure dµ(x). Lemma 2.3. We have (2.4) Proof. y e 2 (y+y ) 2 γ y e 2 (y+y ) 2 (y + y )e 2 (y+y ) 2. γ ( y γ γ exp 2 (y γ + y γ )γ e 2 (y+y ) 2 ) (y γ + y γ )(y γ + y γ ) γ (y + y ) e 2 (y+y ) 2.

6 6 MOTOHICO MULASE AND JOSEPHINE YU The application of powers of (/y) to e 2 (y+y ) 2 produces zero contribution unless two differentiations are paired because of the restriction y 0 at the end. Therefore (2.3) counts the number of pairs of differential operators in the product. Since the trace is invariant under cyclic permutations let us represent each (/y) as a -valent vertex of a ribbon graph. Two vertices are connected if the differentiations from the vertices are paired one another. Since y is killed by the differentiation and (2.5) y y γ γ the edge connecting the vertices comes with an assignment of a group element γ and γ on the two half-edges. With a factor / v! v the integral (2.9) is equal to t v Γ ribbon graph v (Γ)v Aut R Γ µ Γ(G) where µ Γ (G) is the number of assignments of group elements to each half-edge of a ribbon graph Γ subect to the following conditions: Condition. If half-edges E + and E form an edge E of Γ and a group element w is assigned to E + then w is assigned to E ; Condition 2. At every vertex the product of all group elements assigned to half-edges incident to the vertex according to the cyclic order of the vertex is equal to. The second condition comes from the fact w w 2 w w w 2 w 0 oherwise that appears as a -valent vertex of (2.5). Lemma 2.4. The quantity µ Γ (G) is a topological invariant of a compact oriented surface with a fixed number of marked points (or the faces of its cell-decomposition). Proof. This follows from the invariance of µ Γ (G) under an edge contraction and edge insertion [3]. When an edge that is not a loop is contracted the configuration of group elements on the new graph still satisfies Conditions and 2. If the edge is inserted back then we know exactly what group element has to be assigned to each half-edge due to Condition 2. This proves the lemma. Figure 2.. A standard graph for a closed oriented surface of genus g with f marked points or faces. It has f tadpoles on the left and g bi-petal flowers on the right.

7 HOMOMORPHISMS FROM A SURFACE GROUP INTO A FINITE GROUP 7 As in [3] we can use a standard graph for each topology to calculate the number µ Γ (G). If we use Figure 2. as our standard graph Γ then we immediately see that the number of configurations of group elements on this graph is f(γ) Hom(π (S Γ ) G) where the tadpoles of the graph have the contribution of f(γ) in the computation. This completes the proof of the expansion formula (2.8) and by adusting the constant factor in the exponential function of the integrand we establish Theorem 2.. Note that we have a -algebra isomorphism (2.6) C[G] λ Ĝ End(V λ ) which decomposes the character of the regular representation into the sum of irreducible characters: χ reg (dim V λ )χ λ N λ tr Vλ λ Ĝ λ Ĝ where N λ dim V λ and χ λ is its character. Therefore ( log exp ) 2 χ reg(x 2 ) exp t χ reg(x ) dµ(x) 3 ( log exp N ) λ 2 tr V λ (x 2 t ) exp N λ tr V λ (x ) dµ λ (x) λ Ĝ 3 (2.7) ( log exp N ) λ H Nλ 2 tr V λ (x 2 t ) exp N λ tr V λ (x ) dµ λ (x) C λ Ĝ 3 V λ ) Aut R Γ λ Ĝ(dim χ (SΓ) t v (Γ) 3 Γ connected ribbon graph with valence 3 where dµ λ is the normalized Lebesgue measure on the space of N λ N λ hermitian matrices. Comparing the two expressions (2.7) and (2.7) we obtain Eqn.(.). 3. Non-orientable Case For the non-orientable case the prototype integral is a matrix integral over N N real symmetric matrices [7 4]: (3.) log e 4 NtrX2 e N t 2 trx dµ(x) H NR AutΓ Nχ (S Γ ) t v (Γ) Γ connected Möbius graph where what we call a Möbius graph is a graph drawn on a closed surface orientable or non-orientable defining a cell-decomposition of the surface. Its automorphism is an automorphism of the cell-decomposition of the surface that is determined by the graph but this time we allow orientation-reversing automorphisms. Every compact non-orientable surface without boundary is obtained by removing k disoint disks from a sphere S 2 and glue k cross-caps back into the holes. The number of cross-caps is called the cross-cap genus of the non-orientable surface. If S Γ is non-orientable then its cross-cap genus k is determined by χ(s Γ ) 2 k v(γ) e(γ) + f(γ)

8 8 MOTOHICO MULASE AND JOSEPHINE YU where again by f(γ) we denote the number of disoint open disks in S Γ \ Γ. The space H NR of N N real symmetric matrices is a real vector space of dimension N(N + )/2 and dµ(x) is the normalized Lebesgue measure of this space. We note that the coefficients of the integral in (3.) are different from (2.) reflecting the fact that a dihedral group naturally acts on a vertex of a Möbius graph. We can generalize the matrix integral (3.) to an integral over the real group algebra R[G] which is a -algebra with χ reg as a trace function. Theorem 3.. (3.2) log H R[G] e χ 4 reg(x 2) e t χ 2 reg(x ) dµ(x) Γ connected Möbius graph AutΓ χ (S Γ ) Hom(π (S Γ ) G) t v (Γ) where the integral is taken over the space of self-adoint elements of R[G] and S Γ is the orientable or non-orientable surface determined by a Möbius graph Γ. For the proof of this expansion formula we refer to [32]. Recall that the real group algebra R[G] decomposes into simple factors according to the three types of irreducible representations (.2). First we note that Ĝ consists of complex irreducible representations of G that are defined over R. A representation in Ĝ2 is not defined over R and its character is not real-valued. Thus the complex conugation acts on the set Ĝ2 without fixed points. Let Ĝ2+ denote a half of Ĝ2 such that (3.3) Ĝ 2+ Ĝ2+ Ĝ2. Finally a complex irreducible representation of G that belongs to Ĝ3 admits a skewsymmetric bilinear form. In particular its dimension (over C) is even. Now we have a -algebra isomorphism (3.4) R[G] End R (Vλ R ) End C (V λ ) End H (Vλ H ) λ Ĝ λ Ĝ2+ λ Ĝ3 where Vλ R is a real irreducible representation of G that satisfies V λ Vλ R R C. The space Vλ H is a (dim C V λ )/2-dimensional vector space defined over H such that its image under the natural inection (3.5) End H (V H λ ) End C(V λ ) coincides with the image of ρ λ : R[G] End C (V λ ) where ρ λ is the representation of G corresponding to λ Ĝ3. The inective algebra homomorphism Eqn.(3.5) is defined by ( ( ) ( ( ) i i i k. ) i) i The algebra isomorphism Eqn.(3.4) gives a formula for the character of the regular representation on R[G]: (3.6) χ reg (dim R Vλ R ) χ λ + (dim C V λ )(χ λ + χ λ ) + 2(dim C V λ ) trace V H λ λ Ĝ λ Ĝ2+ where in the last term the character is given as the trace of quaternionic (dim C V λ )/2 (dim C V λ )/2 matrices. To carry out the non-commutative integral Eqn.(3.2) we need to λ Ĝ3

9 HOMOMORPHISMS FROM A SURFACE GROUP INTO A FINITE GROUP 9 know the result for quaternionic self-adoint matrix integrals. Fortunately a recent paper [3] provides exactly this necessary formula: (3.7) log e NtrX2 e 2N t trxdµ(x) H NH AutΓ ( 2N)χ (S Γ ) t v (Γ) Γ connected Möbius graph where H NH is the space of N N quaternionic self-adoint matrices. We pay a particular attention to the negative sign in RHS of the formula and the factor 2N. The extra factor 2 cancels with the size of the quaternionic matrices in End H (Vλ H ) which is half of the dimension of V λ. The negative sign in RHS of (3.7) is the source of the negative sign in the second summation term of Eqn.(.3). The computation of the matrix integral of each factor of Eqn.(3.4) then establishes Eqn.(.3). Note that the Ĝ2 component has no contribution in this formula. This is due to the fact that graphical expansion of a complex hermitian matrix integral contains only oriented ribbon graphs. 4. Algebraic Proof of the Counting Formula The counting formulas (.) and (.3) have an algebraic proof [3 24] without going through the computation of matrix integrals. Although the proof is well known to experts in group theory [9] since it is not found in modern textbooks we record it here to illuminate its relation to our graphical expansion formulas. The necessary backgrounds for the algebraic proof are found in Isaacs [7] Serre [37] and Stanley [38]. Let us first identify the group algebra C[G] x } x(γ) γ with the vector space F (G) of functions on G. The convolution product of two functions x(γ) and y(γ) is defined by (x y)(w) x(wγ )y(γ) which makes (F (G) ) an algebra isomorphic to the group algebra. In this isomorphism the set of class functions CF (G) corresponds to the center ZC[G] of C[G]. According to the decomposition into simple factors Eqn.(2.6) we have an algebra isomorphism ZC[G] λ Ĝ C where each factor C is the center of EndV λ. The proection to each factor is given by an algebra homomorphism Now let pr λ : ZC[G] x x(γ) γ pr λ (x) x(γ)χ λ (γ) C. dim V λ (4.) p λ dim V λ χ λ (γ ) γ ZC[G].

10 0 MOTOHICO MULASE AND JOSEPHINE YU The orthogonality of the irreducible characters shows that pr λ (p µ ) δ λµ. Consequently we have p λ p µ δ λµ p λ or more concretely Therefore we have dim V λ χ λ (s ) s dim V µ χ µ (t ) t s G t G ( ) χ λ ((wt ) )χ µ (t ) w dim V λ dim V µ 2 δ λµ dim V λ w G t G χ λ (w ) w. w G χ λ (tw )χ µ (t ) t G or in terms of the convolution product dim V µ δ λµ χ λ (w ) (4.2) χ λ χ µ dim V µ δ λµ χ λ. Now let us consider f g (w) (a b a 2 b 2... a g b g ) G 2g a b a (4.3) b a g b g a g r k (w) (a a 2... a k ) G k a 2 a2 2 a2 k w}. b g w} Since these are class functions on the group G they can be written as linear combinations of the irreducible characters χ λ λ Ĝ} of G. Theorem 4.. For every g and w G we have ( ) 2g χ λ (w). (4.4) f g (w) λ Ĝ dim C V λ Corresponding to the non-orientable case for every k we have (4.5) r k (w) ( ) k χ λ (w) ( ) k χ λ (w). dim C V λ dim C V λ λ Ĝ Proof. From the definition (4.3) we see that In particular λ Ĝ3 f g +g 2 f g f g2 and r k +k 2 r k r k2. f g g-times }} k-times }} f f and r k r r. The general formulas now follow from the formulas for f and r that are found in [38] and [7] respectively using (4.2). Evaluating Eqns.(4.4) and (4.5) at w we obtain the counting formulas (.) and (.3). The quantity g in the formula is of course the genus of an oriented surface and k is the cross-cap genus of a non-orientable surface. The class function f g (resp. r k ) gives the number Hom(π (S) G) when evaluated at w for an oriented (resp. non-orientable) surface S.

11 HOMOMORPHISMS FROM A SURFACE GROUP INTO A FINITE GROUP 5. Remarks Hermitian matrix integrals have been used effectively in the study of topology of moduli spaces of Riemann surfaces with marked points [ ]. These works rely on a relation between graphs and Riemann surfaces (cf. [39]). There is a burst of developments in this direction lately [ ]. Real symmetric matrix integrals are used for the study of moduli spaces of real algebraic curves [4]. In a context of finite groups a striking relation between random matrices and representation theory of symmetric groups is discovered in [ ]. We have seen in this article that an extension of these matrix integrals shows yet another interesting relation between surface geometry and finite group theory. There is a completely different proof of the counting formula Theorem. due to Freed and Quinn []. They use Chern-Simons gauge theory with a finite gauge group. To establish the formula they construct a Chern-Simons gauge theory on each Riemann surface S. Interestingly our approach does not start with a specific manifold which is usually the space-time in physics. In a sense our non-commutative integral is a quantum field theory on a finite group without space-time and surfaces appear in its Feynman diagram expansion. Yet the result shows that Chern-Simons gauge theory on a surface is mathematically equivalent to our non-commutative integrals over group algebras. To be more precise the group algebra model is a generating function of Chern-Simons gauge theory with finite gauge group for all closed surfaces. The original algebraic proofs of the formulas using the convolution product found in [3 24] are indeed related to the cut-and-paste construction of topological quantum field theory that reduces the construction of an invariant on a surface of genus g to surfaces of lower genera [ 4]. A more recent development on counting formulas in finite groups is found in [23]. Acknowledgement. The authors thank Andrei Okounkov for drawing their attention to many classical literatures of the subect. They also thank Dmitry Fuchs Greg Kuperberg Anne Schilling Albert Schwarz Bill Thurston and Andrew Waldron for many valuable comments and stimulating discussions on the subects of this paper. References [] Jinho Baik Percy Deift and Kurt Johansson On the distribution of the length of the longest increasing subsequence of random permutations Journal of American Mathematical Society 2 (999) [2] Jinho Baik Percy Deift and Kurt Johansson On the distribution of the length of the second row of a Young diagram under Plancherel measure math.co/9908 (999). [3] Jinho Baik and Eric Rains Symmetrized random permutations math.co/99009 (999). [4] D. Bessis C. Itzykson and J. B. Zuber Quantum field theory techniques in graphical enumeration Advanced in Applied Mathematics (980) [5] Pavel M. Bleher and Alexander R. Its Random matrix models and their applications Mathematical Sciences Research Institute Publications 40 Cambridge University Press 200. [6] Alexander Borodin Andrei Okounkov and Grigori Olshanski On asymptotics of Plancherel measures for symmetric groups math.co/ (999). [7] C. Brézin C. Itzykson G. Parisi and J.-B. Zuber Planar diagrams Communications in Mathematical Physics 59 (978) [8] William Burnside Theory of groups of finite order Second Edition Cambridge University Press 99. [9] Percy Deift Integrable systems and combinatorial theory Notices of AMS 47 (2000) [0] Richard P. Feynman Space-time approach to quantum electrodynamics Physical Review 76 (949) [] Daniel S. Freed and Frank Quinn Chern-Simons theory with finite gauge group Communications in Mathematical Physics 56 (993) [2] Georg Frobenius Über Gruppencharaktere Sitzungsberichte der königlich preussischen Akademie der Wissenschaften (896) [3] Georg Frobenius and Isaai Schur Über die reellen Darstellungen der endlichen Gruppen Sitzungsberichte der königlich preussischen Akademie der Wissenschaften (906)

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