Explicit Formulas for 2-Characters

EXPLICIT F ORMULAS F OR 2-CHARACTERS ANG ´ ELICA M. OSORNO Abstract. Gan ter and Kaprano v asso ciated a 2-cha racter to 2-r epresentations of a finite group. Elgueta classified 2-r epresen tations in the category of 2-v ector spaces 2 V ect k in terms of cohomological data. W e give an explicit formula for the 2- c haracter i n terms of this cohomological data and derive some conse- quences. 1. Introduction In [4 ], Hopkins, Kuhn and Rav enel develop a theory of genera lized characters tha t computes E ∗ ( B G ) for the n -th Mor a v a E -theory . The c haracters in this cas e are class functions defined o n the set of n -tuples of commuting elemen ts of G whose order is a p ow er of p . In [3], Ganter a nd Kapranov define a 2-character for a 2- representation of a finite gro up in a 2- c ategory . This 2-character is a function that assigns an elemen t of the fie ld k to every pair ( g, h ) of c omm uting elements in G . Ganter and Kapra no v prov ed tha t these 2- c haracter s sa tisfy the sa me formulas as the characters in [4] for n = 2. The purp ose of this pa per is to find an explicit descr iption of the 2- c haracter s of a 2-repre s en tation in the 2-categ ory of 2-vector spaces, 2 V ect k . In order to find this description, we fir st re v iew the alg ebraic cla ssification o f 2-r epresentations. In [2], it is shown that every equiv alence cla s s of 2- representations is given uniquely by a finite G -s et S and a class in H 2 ( G ; k S ). W e present a streamlined a pproach to this result. W e then p ro ceed to compute the 2-character in ter ms of this associa ted coho mol- ogy class. Using these computations w e prov e that 2 - c haracter s a re additiv e and m ultiplicative with resp ect to direct sum and tens o r pr oduct of 2 -representations. Given a 2-r epresent ation ρ of H ⊆ G , Ganter a nd K apranov als o define the induced representation, a nd compute its character in terms o f the character of ρ . Using o ur cohomolog ical classifica tion of representations, w e identify the induced repres e n ta- tion in ter ms of the cohomolog ical data fo r ρ using the Shapiro isomor phis m. Finally , us ing the explicit computation o f 2-characters we give a n example of t wo non-equiv alen t 2 - representations that ha ve the same character, thus showing that this assignment is no t faithful. The 2-categ o rical language can b e cumbersome, so we rev iew some o f the terminol- ogy and constructions. F or mo re background on 2- categories w e r efer the reader to [3] and [2 ]. The author would like to thank Nora Ganter for many helpful discussio ns that lead to some of t he a pproaches and r esults pr esen ted here. The author w ould a lso like to thank Mark Behrens for his comments o n e a rlier versions o f this pap er. 1 2. 2-represent a tions and their characters F ollowing [3], we review the notions o f 2- r epresentations of a gr oup a nd ch ara cter theory . Definition 1. Let C be a 2- category and G a group. A 2-r epr esentation of G in C is a la x 2-functor fr om G (viewed as a discrete 2- category) to C . This amounts to the following data: (1) an ob ject V of C , (2) for every g ∈ G , a 1-mo rphism ρ g : V → V , (3) a 2-is omorphism φ 1 : ρ 1 ⇒ 1 C , (4) for every pair g , h ∈ G , a 2-isomo rphism φ g,h : ρ g ◦ ρ h ⇒ ρ gh . This da ta has to satisfy the following c o nditions: (5) (asso ciativit y) for every g , h , k ∈ G , φ ( gh, k ) ( φ g,h ◦ ρ k ) = φ ( g,h k ) ( ρ g ◦ φ h,k ) , (6) for any g ∈ G , φ 1 ,g = φ 1 ◦ ρ g and φ g, 1 = ρ g ◦ φ 1 . There is a 2 -category in which we are particular ly interested: the 2- category of 2-vector spaces. There are several 2-ca tegories which ar e 2-eq uiv alent and give equiv a len t 2-repr esen tation theories. W e will use the definition in [5]. Definition 2. L e t k be a field. The 2-category 2 V ect k has as ob jects [ n ], wher e n ∈ { 0 , 1 , 2 , . . . } . F or in tegers m, n , the set of 1-morphisms 1 H om 2 V ect k ([ m ] , [ n ]) is the s e t of m × n matrices with en tries in k -vector spaces. These are ca lled 2- matrices. Co mposition is given b y ma trix mu ltiplication using tensor pro duct and direct sum. F or 2-matrices A and B of the same size, 2-morphisms are given b y matrices of linear maps φ , with φ ij : A ij → B ij . Thu s a 2-repr e s en tation o f a gro up G in 2 V ect k consists of the following data: (1) A natural num b er n , c a lled the dimension, (2) for every g ∈ G , an n × n 2- matrix, ρ g (3) a 2-is omorphism φ 1 : ρ 1 ⇒ 1 [ n ] , (4) for every pair g , h ∈ G , a 2-isomo rphism φ g,h : ρ g ◦ ρ h ⇒ ρ gh . The isomo rphisms φ g,h and φ 1 are sub ject to the same c o nditions express ed a bov e. The following approach is similar to a more general re s ult in [2]. The r esult is presented in a co or dina te-free appro ac h. Prop osition 1. Ther e is a one-to-one c orr esp ondenc e b etwe en e quivalenc e classes of 2-r epr esentations of G in 2 V ect C and p airs ( S, [ c ]) wher e S is a finite G -set and [ c ] ∈ H 2 ( G ; ( C × ) S ) . Her e ( C × ) S denotes ( C × ) | S | as a G -mo dule thr ough the action of G on S . Pr o of. Note that since ρ g ρ g − 1 is is omorphic to 1 [ n ] , each ρ g is given b y a weakly in- vertible 2-matrix. This means that the entries in ρ g can only be 0 and 1-dimensiona l vector spaces, w ith ex a ctly o ne entry per row and co lumn b eing 1-dimensiona l. That is, up to isomorphis m, ρ g is given by an n × n p e rm utation matrix. Th us, we ca n think of ρ as a map 2 ρ : G → Σ n . Now, let us turn our attent ion t o φ g,h and φ 1 . The 2-matrices ρ gh and ρ g ρ h hav e only one nonzero entry p er row and column, and those entries are 1-dimensional vector space s. Thu s, to sp ecify the 2-iso morphism φ g,h all we need to g iv e is a sequence of n nonzer o co mplex num bers { c i ( g , h ) } which give the isomor phism fo r the nonzer o entry in the i th row. Condition (5) in the definition o f a 2-repr esen tation implies c σ − 1 ( i ) ( h, k ) · c i ( g , hk ) = c i ( g h, k ) · c i ( g , h ) , where σ is the p ermut ation represe n ted by ρ g . W e can think of ( C × ) n as a G -mo dule through ρ , where g · − → a = ρ g − → a in ma trix notation. W e will denote this G -mo dule by ( C × ) n ρ . W e can then think of c a s a 2 -cochain G × G → ( C × ) n ρ . Then the condition ab o ve bec omes the co cycle co ndition: ( δ c )( g , h, k ) = g · c ( h, k ) − c ( g h, k ) + c ( g , h k ) − c ( g , h ) = 0 Here w e are using additive nota tion f or the compo nen t wise multiplication group structure of C × ) n . On the other ha nd, Co nditio n (6) of Definition 1 with g = 1 implies that φ 1 is given by multiplication by c (1 , 1). Hence, we can sa y that up to isomorphism, a 2-representation is determined by a group ho momorphism ρ : G → Σ n and a 2-co cycle c ∈ C 2 ( G ; ( C × ) n ρ ). This coincides with the notio n in [2]. In this new langua ge, we would like to identify the equiv alence class e s of repr esen ta- tions. Two representations are equiv alent if there exis ts a ps eudonatural equiv a le nc e betw een the vectors. A pseudonatur a l transformation is a 1-mo r phism f : [ n ] → [ n ′ ] together with a 2- is omorphism ψ ( g ) : ρ ′ g ◦ f ⇒ f ◦ ρ g for every g ∈ G , sa tisfying t wo co her ence conditions: (1) F or all g , h ∈ G , ψ ( g h ) · ( φ ′ g,h ◦ 1 f ) = ( 1 f ◦ φ g,h ) · ( ψ ( g ) ◦ 1 ρ h ) · ( 1 ρ ′ g ◦ ψ ( h )), (2) φ ′ 1 ◦ 1 f = ( 1 f ◦ φ 1 ) · ψ (1 ). This pseudona tural transformation is a n equiv alence if and only if f is a weakly inv ertible 1-morphism, that is if n = n ′ and f is given by a p ermutation matrix. Assume t w o 2-representations a re eq uiv alent. If these 2-representations are giv en by the same map ρ : G → Σ n and f = 1 [ n ] , the 2 -isomorphism ψ ( g ) is given by a sequence of n nonzero co mplex n umbers b i ( g ) which give the isomorphisms on the nonzer o 1- dimensional v ector spaces in ea c h row. Ag ain, we can think of these vectors of complex n um b ers as a 1-co c hain G → ( C × ) n . The tw o coher ence conditions imply for all i : b i ( g h ) c ′ i ( g , h ) = c i ( g , h ) b i ( g ) b σ − 1 ( i ) ( h ) , where c and c ′ are the c o cycles giving the tw o repre s en tations. If we write this in additive nota tion w e get: 3 ( δ b )( g , h ) = g · b ( h ) − b ( g h ) + b ( g ) = c ′ ( g , h ) − c ( g , h ) . That is, c and c ′ are c o homologous co cycles in C 2 ( G ; ( C × ) n ρ ) if and only if they give equiv alent r epresentations. In gener al the re pr esen tations given by ρ , [ c ] and ρ ′ , [ c ′ ] ar e equiv a len t if and only if there exists a p erm utation f ∈ Σ n such that ρ ′ g = f ρ g f − 1 and [ c ′ ] = [ f · c ]. This follows from the asse r tions a bov e. This proves the theorem.  3. Direct sum and tensor product There is a notion o f direct sum a nd tensor pro duct in 2 V ect k as noted in [5]. Direct sum is given a s follows: • On ob jects: [ n ] ⊕ [ m ] = [ n + m ], • on 1- morphisms is given b y blo ck sum of 2-matrices, • on 2- morphisms is given b y blo ck sum of matrices of linear maps. T ensor pro duct is g iv en as fo llo ws: • On ob jects: [ n ] ⊗ [ n ′ ] = [ nn ′ ]. • on 1-morphisms : let f : [ m ] → [ n ], f ′ : [ m ′ ] → [ n ′ ] be 1-mor phisms, then f ⊗ f ′ : [ mm ′ ] → [ nn ′ ] is the 2 -matrix with ( i, i ′ ) , ( j, j ′ )-entry equal to f ij ⊗ f ′ i ′ j ′ , where the set of mm ′ elements is lab eled by pa irs ( i , i ′ ), wher e i = 1 , . . . , m , i ′ = 1 , . . . , m ′ , with the order: (1 , 1) , (1 , 2) , . . . , (1 , n ′ ) , (2 , 1) , . . . , ( m, m ′ ) and simila rly fo r nn ′ . • on 2- morphisms: s imilarly as a bov e. These ope rations can b e extended to 2-repr esen tations on 2 V ect k by taking the ap- propriate dir ect sum a nd/or tensor pro duct of the resp ectiv e o b jects, 1-morphisms and 2-morphisms. It is not hard to prov e that we obtain a new 2-re pr esen tation in bo th cases. 4. Induced 2-R epresent a tions In [3], Gan ter and Kapranov define the notion of a n induced representation given H ⊆ G and inclusion o f finite groups . Here we ana lyze the case of 2 V ect k following their explicit description in Remark 7.2. Let ρ : H → Σ n , [ c ] ∈ H 2 ( H ; ( C × ) n ρ ) be a 2-repres e n tation on [ n ]. Let S b e the corres p onding H -s et. Let m b e the index o f H in G . Let R = { r 1 , . . . , r m } be a system of representatives of the left cose ts of H in G . Then ind | G H ( ρ ) is a 2-r epresentation of G of dimension nm . The matrix for ind | G H ρ g is a blo c k matrix, with blo cks o f size n × n , wher e the ( i, j )- th block is given as follows: (ind | G H ρ g ) ij = ( ρ h if g r j = r i h, h ∈ H 0 else. . Now we turn our a tten tion to ind | G H φ g 1 ,g 2 . Notice that 4 (ind | G H ρ g 1 ) ◦ 1 (ind | G H ρ g 2 ) ik = = ( ρ h 1 ◦ 1 ρ h 2 if g 1 r j = r i h 1 and g 2 r k = r j h 2 0 else. and the ( i , k )-th block is not zero prec is ely when since g 1 g 2 r k = r i h 1 h 2 , that is, when (ind | G H ρ ( g 1 g 2 ) ) ik = ρ ( h 1 h 2 ) . Here ◦ 1 denotes vertical comp osition o f 2-mor phisms. On this blo c k then (ind | G H φ g 1 ,g 2 ) ik = φ h 1 ,h 2 . Notice that the G -set given by ind | G H ρ : G → Σ nm is pr e cisely ind | G H = G × H S ∼ = R × S . Thus, the G -mo dule ( C × ) ind | G H S is ind | G H [( C × ) S ]. The corr esponding co cycle is then (ind | G H c ) ( r i ,s ) ( g 1 , g 2 ) = c s ( h 1 , h 2 ) , where ( r i , s ) ∈ R × S and h 1 , h 2 are as ab ov e. One can tr ace that [ind | G H c ] is the image of [ c ] under the Shapiro isomo rphism H 2 ( H ; ( C × ) S ) ∼ = H 2 ( G ; ind | G H [( C × ) S ]) ∼ = H 2 ( G ; ( C × ) ind | G H S ) . In particular , le t ( S, [ c ]) be an e quiv alence c la ss of a representation of G . Let S = k a i =1 G/H i be the decomp osition of S into G orbits. Then the chain of iso mo rphisms H 2 ( G ; ( C × ) S ) ∼ = k M i =1 H 2 ( G ; ( C × ) G/H i ) ∼ = k M i =1 H 2 ( H i ; ( C × )) sends [ c ] to [ c 1 ] ⊕ · · · ⊕ [ c k ] to [ d 1 ] ⊕ · · · ⊕ [ d k ], where [ d i ] is the imag e of [ c i ] under the Shapiro is omorphism. This means that the re presen tation giv en by ( G/H i , [ c i ]) is the induced r e presen- tation of a 1-dimensio nal r epresent ation ( ∗ , [ d i ]) of H i W e thus r eco ver the following pr opos ition, which coincides with [3, P rop. 7.3]. Prop osition 2. Every r epr esentation is the dir e ct sum of induc e d 1-dimensional r epr esentations. 5 5. 2-chara cters W e w ould like kno w what the character introduced in [3] looks like in terms of ρ and c . Definition 3. Giv en a 2-ca teg ory C , let A be an ob ject and F : A → A a 1- endomorphism. W e define the c ate gori c al t r ac e as T r ( F ) = 2 H om C ( 1 A , F ) . Note t hat since C is a 2-catego ry , E nd ( A ) = H om ( A , A ) is a category . This defi- nition gives a functor T r : E nd ( A ) → S et . If α : F ⇒ G is a 2-morphism b et ween F, G ∈ E nd ( A ), T r ( α ) : 2 H om C ( 1 A , F ) → 2 H om C ( 1 A , G ) is given b y co mp osition with α . Let D b e a category . W e recall that a 2-categ o ry is enriched ov er D if the categor ie s H om ( A, B ) are enriched ov er D for all o b jects A, B . If the 2-catego ry C is enriched ov er D then T r is a functor into D . When C = V ect k , A = [ n ] and F is an n × n matrix [ F ij ] of v ector spa c e s. Then the following equality ho lds T r ( F ) = n M i =1 F ii . Note that the categor ical trace is additive and multiplicative in the following sense. Let F : [ n ] → [ n ] and G : [ m ] → [ m ]. Then F ⊕ G : [ n + m ] → [ n + m ] and T r ( F ⊕ G ) = n + m M i =1 ( F ⊕ G ) ii = ( n M i =1 F ii ) ⊕ ( m M i =1 G ii ) = T r ( F ) ⊕ T r ( G ) . Also, F ⊗ G : [ n · m ] → [ n · m ] and T r ( F ⊗ G ) = M i =1 ,...,n j =1 ,...,m ( F ⊗ G ) ( i,j )( i,j ) = M i =1 ,...,n j =1 ,...,m F ii ⊗ G j j = ( n M i =1 F ii ) ⊗ ( m M i =1 G j j ) = T r ( F ) ⊗ T r ( F ) . Given F : A → A , let G : A → B b e an equiv alence with qua si-in verse H . Then the trac e is conjugation in v ariant in the sense that there is an iso morphism: ψ : T r ( F ) → T r ( GF H ) . Let ρ be a 2-repre s en tation of a group G in C . The character of ρ is given b y assigning to each g ∈ G the trace T r ( ρ g ). In this case, since we hav e the maps φ g,h and φ 1 we can use the conjugation inv ar iance above to get a map ψ g ( h ) : T r ( ρ h ) → T r ( ρ ghg − 1 ) . 6 When ρ is a 2-r epresentation in 2 V ect k and g a nd h co mm ute, T r ( ρ h ) and T r ( ρ ghg − 1 ) are the same vector space. Th us Ganter and Kapranov introduce the following def- inition: Definition 4. The 2-char acter of ρ is a function on pair s o f co mm uting elements: χ ρ ( h, g ) = tr ace  ψ g ( h ) : T r ( ρ h ) → T r ( ρ h )  . It is inv ar ian t under s imultaneous conjuga tion. Note that since T r , ψ g ( h ), and tr ace are all additive and m ultiplicative, we deduce that the character is also additive and multiplicativ e. (W e will later g iv e a different, explicit pro of of this fact). W e are now prepared to compute the character of a 2-repres e ntation on 2 V ect C . Theorem 1. The c ate goric al char acter of a 2-r epr esentation given by ρ : G → Σ n and c ∈ C 2 ( G ; ( C × ) n ρ ) is χ ( g , h ) = X i = ρ g ( i )= ρ h ( i ) c i ( g , g − 1 ) − 1 c i (1 , 1) − 1 c i ( h, g − 1 ) c i ( g , hg − 1 ) . Pr o of. Let ϕ : 1 [ n ] ⇒ ρ h , that is, a vector in T r ( ρ h ). No te that this is an n × n matrix with zero entries everywhere except in those diag onal entries that a re nonzero in ρ h . With out lo ss of generality , we can assume those are in the fir st k rows (w e can co njuga te ρ b y a p erm utation matrix f and change c according ly to f · c ; this will just give a reorder ing of the indices which does not change the dimension of T r ( ρ h ) nor the map ψ g ( h )). Thus w e have ϕ =           a 1 · · · 0 0 · · · 0 . . . . . . . . . . . . . . . . . . 0 · · · a k 0 · · · 0 0 · · · 0 0 · · · 0 . . . . . . . . . . . . . . . . . . 0 · · · 0 0 · · · 0           , where k is the num ber o f indices fixed by ρ h . W e would lik e to c ompute now ψ g ( h ) = φ g,h , g − 1 · ( ρ g ◦ ϕ ◦ ρ g − 1 ) · φ − 1 g,g − 1 · φ − 1 1 for commuting pa irs ( g , h ). Note that since g and h commute, ρ h and ρ g ◦ ρ h ◦ ρ g − 1 are isomorphic, in par ticular, the nonzero en tries in the diagona l are in the same po sition. Th us we hav e 7 ρ g ◦ ϕ ◦ ρ g − 1 =           a ρ g − 1 (1) · · · 0 0 · · · 0 . . . . . . . . . . . . . . . . . . 0 · · · a ρ g − 1 ( k ) 0 · · · 0 0 · · · 0 0 · · · 0 . . . . . . . . . . . . . . . . . . 0 · · · 0 0 · · · 0           . On the other hand, comp osition with the iso mo rphisms φ is given just by multipli- cation by the appro priate scalar in the appro priate row: ψ g ( h ) = φ g,h , g − 1 · ( ρ g ◦ ϕ ◦ ρ g − 1 ) · φ − 1 g,g − 1 · φ − 1 1 =           b 1 · · · 0 0 · · · 0 . . . . . . . . . . . . . . . . . . 0 · · · b k 0 · · · 0 0 · · · 0 0 · · · 0 . . . . . . . . . . . . . . . . . . 0 · · · 0 0 · · · 0           , where b i = c i ( g , g − 1 ) − 1 c i (1 , 1) − 1 c i ( h, g − 1 ) c i ( g , hg − 1 ) a ρ − 1 g ( i ) . W e can think of the matrices { e i } k i =1 , where e i is th e n × n matrix with 1 in the ( i, i ) entry and zer o everywhere e lse, as a basis for T r ( ρ h ). Then the contribution to the character co mes from the indices i fix ed b y b oth ρ h and ρ g : χ ( g , h ) = X i = ρ g ( i )= ρ h ( i ) c i ( g , g − 1 ) − 1 c i (1 , 1) − 1 c i ( h, g − 1 ) c i ( g , hg − 1 ) .  Lemma 1. The ch ar acter is invariant u nder e quival enc e. Pr o of. Given tw o equiv alen t representations given by ρ , [ c ] ∈ H 2 ( G ; ( C × ) n ρ ) and ρ ′ , [ c ′ ] ∈ H 2 ( G ; ( C × ) n ρ ′ ), there exists f ∈ Σ n such that ρ ′ = f − 1 ρf a nd [ c ′ ] = [ f · c ], with ( δ b )( g , h ) = g · b ( h ) − b ( g h ) + b ( g ) = c ′ ( g , h ) − f · c ( g , h ) . Then 8 χ ′ ( g , h ) = X i = ρ ′ g ( i )= ρ ′ h ( i ) c ′ i ( g , g − 1 ) − 1 c ′ i (1 , 1) − 1 c ′ i ( h, g − 1 ) c ′ i ( g , hg − 1 ) = X i = f ρ g f − 1 ( i )= f ρ h f − 1 ( i ) b i (1) c f − 1 ( i ) ( g , g − 1 ) b i ( g − 1 ) b i ( g ) · b i (1) c f − 1 ( i ) (1 , 1) b i (1) b i (1) · c f − 1 ( i ) ( h, g − 1 ) b i ( g − 1 ) b i ( h ) b i ( hg − 1 ) · c f − 1 ( i ) ( g , hg − 1 ) b i ( hg − 1 ) b i ( g ) b i ( h ) = X i = ρ g ( i )= ρ h ( i ) c i ( g , g − 1 ) − 1 c i (1 , 1) − 1 c i ( h, g − 1 ) c i ( g , hg − 1 ) = χ ( g , h ) .  Lemma 2. The char acter re sp e cts the additive and mu ltiplic ative s tructur e of r ep- r esentations. Pr o of. Let ρ, [ c ] and ρ ′ , [ c ′ ] represe nt tw o repres en tations of dimensions n , n ′ .. The direct s um repr esen tation is given b y ˜ ρ h , the blo c k s um o f the matrices ρ h and ρ ′ h for every h and the co cycle ˜ c with ˜ c i = ( c i if i ≤ n, c ′ i − n if i > n. The character of the dir ect sum is ˜ χ ( g , h ) = X i = ˜ ρ g ( i )= ˜ ρ h ( i ) ˜ c i ( g , g − 1 ) − 1 ˜ c i (1 , 1) − 1 ˜ c i ( h, g − 1 )˜ c i ( g , hg − 1 ) = X n ≥ i = ρ g ( i )= ρ h ( i ) c i ( g , g − 1 ) − 1 c i (1 , 1) − 1 c i ( h, g − 1 ) c i ( g , g h − 1 ) + X i>n i − n = ρ ′ g ( i − n )= ρ ′ h ( i − n ) c ′ i − n ( g , g − 1 ) − 1 c ′ i − n (1 , 1) − 1 c ′ i − n ( h, g − 1 ) c ′ i − n ( g , hg − 1 ) = χ ( g , h ) + χ ′ ( g , h ) . On the other ha nd, let ρ, [ c ] denote the tensor pr o duct of ρ, [ c ] and ρ ′ , [ c ′ ]. F rom the definition o f the tens or pr oduct a nd us ing the lab eling ab o ve for the set of nn ′ elements, it is no t ha rd to see that ( ρ g ) ( i,i ′ ) , ( j,j ′ ) = ( ρ g ) i,j ( ρ ′ g ) i ′ ,j ′ , c ( i.i ′ ) = c i c ′ i ′ . Then ( i, i ′ ) is fixed by ρ h if and only if i is fixed by ρ h and i ′ is fixed b y ρ ′ h . Thu s 9 χ ( g , h ) = X ( i,i ′ )= ρ g ( i,i ′ )= ρ h ( i,i ′ ) c ( i,i ′ ) ( g , g − 1 ) − 1 c ( i,i ′ ) (1 , 1) − 1 c ( i,i ′ ) ( h, g − 1 ) c ( i,i ′ ) ( g , hg − 1 ) = X ( i,i ′ ) i = ρ g ( i )= ρ h ( i ) i ′ = ρ g ( i ′ )= ρ h ( i ′ ) (( c i c ′ i ′ )( g , g − 1 )( c i c ′ i ′ )(1 , 1)) − 1 ( c i c ′ i ′ )( h, g − 1 )( c i c ′ i ′ )( g , hg − 1 ) =  X i = ρ g ( i )= ρ h ( i ) c i ( g , g − 1 ) − 1 c i (1 , 1) − 1 c i ( h, g − 1 ) c i ( g , hg − 1 )  ·  X i ′ = ρ ′ g ( i ′ )= ρ ′ h ( i ′ ) c ′ i ′ ( g , g − 1 ) − 1 c ′ i ′ (1 , 1) − 1 c ′ i ′ ( h, g − 1 ) c ′ i ′ ( g , hg − 1 )  = χ ( g , h ) χ ′ ( g , h ) . Hence we se e t hat the c haracters resp ect the additive a n m ultiplicative structures on the representations.  R emark 1 . One can a lso use Theor em 1 to repro duce the formula for the c hara cter of the induced repre s en tation which appea rs in [3, Co rollary 7.6]: χ ind ( g , h ) = 1 | H | X s − 1 ( g,h ) s ∈ H × H χ ( s − 1 g s, s − 1 hs ) . One might hop e that, analog ous to the case of 1-ch ara cters, the map fro m equiv- alence classes o f 2-r epresentations to characters is injective. This t urns out no t to be true, a s the following counterexample shows. Example 1 . W e will consider tw o 2 - representations of Σ 3 of dimension 8 with trivia l co cycle, so t hey amount to a gro up homomorphism ρ : Σ 3 → Σ 8 , that is, they are per m utation representations. Note also tha t they are isomo rphic are p ermutations representations if and only if they ar e isomorphic as 2-repres en tations. Since they hav e tr ivial co cycle, the character is given by χ ( g , h ) = X i = ρ g ( i )= ρ h ( i ) 1 = # { i = ρ g ( i ) = ρ h ( i ) } . Let ρ b e g iv en by three blo c ks: the reg ular representation (action o f Σ 3 on itself ), and tw o triv ial blo c ks. Let ρ ′ be given b y thr e e blo cks as well: 2 blocks with t he action of Σ 3 on Σ 3 / h (12) i and o ne blo c k with the actio n of Σ 3 on Σ 3 / h (123) i . Note that these tw o 2-represe ntations are not iso mo rphic: Σ 3 fixes the last t wo elements of ρ while Σ 3 fixes no ele ment of ρ ′ . On the other hand, we can prove that these t wo representations hav e the same character: the pairs of commuting elements in Σ 3 are those containing 1 and { (123) , (1 32) } . W e can directly compute the characters: χ (1 , 1) = χ ′ (1 , 1) = 8; χ (1 , g ) = χ ′ (1 , g ) = 2 for a ll g 6 = 1 and χ ((123 ) , (1 32)) = χ ′ ((123) , (132 )) = 2. 10 References [1] M. F. Atiy ah. Characters and cohomology of finite groups. Inst. Ha utes ´ Etudes Sci. Publ. Math. ,9:23-64, 1961. [2] J. Elgueta. Represen tation theory of 2-gr oups on Kaprano v and V o ev odsky’s 2-category 2V ect . arXiv: math.CT/0408120 . [3] N. Gante r and M. Kapranov. Representat ion and cha racter theory in 2-categories. A dv. Math., 217(5):226 8-2300, 2008. [4] M. J. Hopkins, N. J. Kuhn and D. C. Rav enel. Generalized group ch aracters and complex orient ed cohomology theories. J. Amer. Math. So c., 13: 553 -594, 2000. [5] M. Kaprano v and V . V oevodsky . 2-categories and Zamolo dc hiko v tetrahedra equations. Pr o c. Symp os. Pur e Math. , 56:177-260. A merican Mathematical So ciet y , 1994. [6] J.-P . Serr e. R epr´ esentations Li n ´ eair es des Gr oups Finis. Hermann, Paris, 1967. 11