On the arithmetic of integral representations of finite groups - PowerPoint PPT Presentation

On the arithmetic of integral representations of finite groups Dmitry Malinin Department of Mathematics UWI Spa, Belgium, June 23, 2017 Dmitry Malinin On the arithmetic of integral representations

Absolutely irreducible orders R – a Dedekind domain with quotient field K , H – R -order, L – absolutely irreducible H -representation module (i.e., a finitely generated H -module that is torsion free as an R -module and such that K ⊗ R L is an absolutely irreducible K ⊗ R H -module). The most interesting case H = RG , G - a finite group and charK ∤ | G | Jordan-Zassenhaus Theorem : Every isomorphism class of KH-representation modules splits in a finite number of isomorphism classes of RH-representation modules if the ideal class group of R is finite. What happens if cl ( R ) is infinite? Dmitry Malinin On the arithmetic of integral representations

p -groups G Theorem. Let K denote either Q p ( ζ p ∞ ) the extension of Q p obtained by adjoining all roots ζ p i , i = 1 , 2 , 3 , ... of p-primary orders of 1. Let us fix the degree t of matrix representations. Let G be any finite nonabelian p-group, and let O K be the ring of integers of K. Then there is an infinite number of integral pairwise inequivalent absolutely irreducible representations of finite groups G in GL n ( O K ) . The constructed representations are contained in the kernel of reduction I t ( mod P ) modulo some prime divisor P of p. Dmitry Malinin On the arithmetic of integral representations

One combinatorial construction for p-groups Let K be a finite extension of either the rational p -adic field Q p , or the field Q of rationals, and let O K be its ring of integers. Consider a group G 0 generated by two elements a and b of order t = p m , a t = b t =1 such that the commutator c = [ a , b ] is contained in the center of G 0 , and c t =1. Let ζ be a primitive root of 1 of degree t . Dmitry Malinin On the arithmetic of integral representations

A combinatorial construction � n − j � � ( − 1 ) i − j C = e ij , i − j n ≥ i ≥ j ≥ 1 � n − j � � C 1 = e ij . i − j n ≥ i ≥ j ≥ 1 Let X = diag ( 1 , x , x 2 , . . . , x t − 1 ) , then � n − j � � x j − 1 ( 1 − x ) i − j e ij , C 1 XC = i − j n ≥ i ≥ j ≥ 1 and if we take x = 1, this will imply C − 1 = C 1 . Dmitry Malinin On the arithmetic of integral representations

A combinatorial construction � n − j � � ( − 1 ) i − j C = e ij , i − j n ≥ i ≥ j ≥ 1 � n − j � � C 1 = e ij . i − j n ≥ i ≥ j ≥ 1 Let X = diag ( 1 , x , x 2 , . . . , x t − 1 ) , then � n − j � � x j − 1 ( 1 − x ) i − j e ij , C 1 XC = i − j n ≥ i ≥ j ≥ 1 and if we take x = 1, this will imply C − 1 = C 1 . Dmitry Malinin On the arithmetic of integral representations

The construction for 2-generated p-groups The following representation of G 0 is faithful and absolutely irreducible.  . . .  0 1 0 . ... .   . . . . . . .   A = ∆( a ) = ,       0 0 . . . 1   1 0 . . . 0 B = ∆( b ) = diag ( 1 , ζ, . . . , ζ t − 1 ) Dmitry Malinin On the arithmetic of integral representations

The construction for 2-generated p-groups Let X = diag ( 1 , x , x 2 , . . . , x t − 1 ) , then � n − j � C − 1 XC = � x j − 1 ( 1 − x ) i − j e ij , i − j n ≥ i ≥ j ≥ 1 If we take x = ζ, we will obtain: � n − j � ∆ ′ ( b ) = C − 1 ∆( b ) C = C − 1 BC = � ζ j − 1 ( 1 − ζ ) i − j e ij , i − j n ≥ i ≥ j ≥ 1   1 − t 1 0 . . . 0 0 � t � − 1 1 . . . 0 0   2   ... ... ∆ ′ ( a ) = C − 1 AC =   , . . . . . . . . . . . .     � t �  − 0 0 . . . 1 1  t − 1   . . . 0 0 0 0 1 Dmitry Malinin On the arithmetic of integral representations

The construction for 2-generated p-groups Let X = diag ( 1 , x , x 2 , . . . , x t − 1 ) , then � n − j � C − 1 XC = � x j − 1 ( 1 − x ) i − j e ij , i − j n ≥ i ≥ j ≥ 1 If we take x = ζ, we will obtain: � n − j � ∆ ′ ( b ) = C − 1 ∆( b ) C = C − 1 BC = � ζ j − 1 ( 1 − ζ ) i − j e ij , i − j n ≥ i ≥ j ≥ 1   1 − t 1 0 . . . 0 0 � t � − 1 1 . . . 0 0   2   ... ... ∆ ′ ( a ) = C − 1 AC =   , . . . . . . . . . . . .     � t �  − 0 0 . . . 1 1  t − 1   . . . 0 0 0 0 1 Dmitry Malinin On the arithmetic of integral representations

The construction for 2-generated p-groups Consider a finite extension L h = K ( ζ ) of K , its maximal order O L h , a prime divisor P of p and its prime element π h , this prime element may be chosen as ζ p m − 1 in a cyclotomic field K = Q ( ζ p m ) or K = Q p ( ζ p m ) , where ζ p m is a primitive p -root of 1. Let D = diag ( 1 , π h , π 2 h , . . . , π t − 1 ) , then h   1 − t π h 0 . . . 0 0 � t π − 1 � − 1 π h . . . 0 0   2 h   ... ... ∆ h ( a ) = D − 1 h ∆ ′ ( a ) D h =   , . . . . . . . . . . . .     � t π 2 − t �  − 0 0 . . . 1 π h  t − 1 h   . . . 0 0 0 0 1 � n − j � � ζ j − 1 ( 1 − ζ ) i − j π j − i ∆ h ( b ) = D − 1 h ∆ ′ ( b ) D h = h e ij . i − j n ≥ i ≥ j ≥ 1 Dmitry Malinin On the arithmetic of integral representations

The construction for 2-generated p-groups Consider a finite extension L h = K ( ζ ) of K , its maximal order O L h , a prime divisor P of p and its prime element π h , this prime element may be chosen as ζ p m − 1 in a cyclotomic field K = Q ( ζ p m ) or K = Q p ( ζ p m ) , where ζ p m is a primitive p -root of 1. Let D = diag ( 1 , π h , π 2 h , . . . , π t − 1 ) , then h   1 − t π h 0 . . . 0 0 � t π − 1 � − 1 π h . . . 0 0   2 h   ... ... ∆ h ( a ) = D − 1 h ∆ ′ ( a ) D h =   , . . . . . . . . . . . .     � t π 2 − t �  − 0 0 . . . 1 π h  t − 1 h   . . . 0 0 0 0 1 � n − j � � ζ j − 1 ( 1 − ζ ) i − j π j − i ∆ h ( b ) = D − 1 h ∆ ′ ( b ) D h = h e ij . i − j n ≥ i ≥ j ≥ 1 Dmitry Malinin On the arithmetic of integral representations

Globally irreducible representations over arithmetic rings The phenomenon of global irreducibility was first distinguished by J. G. Thompson in the course of constructing the sporadic finite simple group F 3 . An essential ingredient of his construction was an even unimodular lattice Λ 248 of rank 248 with Aut (Λ 248 ) = Z 2 × F 3 . Thompson observed that (Λ 248 , F 3 ) is an example of pairs ( G , Λ) , G is a finite group and Λ is a torsion free Z G -module of finite rank, which satisfy the following condition: Λ / p Λ is an irreducible F p G -module, for every prime p . Many further results and examples are due to Pham Huu Tiep. We are interested in generalizations of the concept of global irreducibility for arithmetic rings introduced by F . Van Oystaeyen and A.E. Zalesskii. Dmitry Malinin On the arithmetic of integral representations

Globally irreducible representations over arithmetic rings Definition 1. We say that M ( n , R ) is a Schur ring if M ( n , R ) = � G � R , the R -span of G , for some finite group G ⊂ GL ( n , R ) . Definition 2. Let F be an algebraic number field and let R be its ring of integers. A subgroup G ⊂ GL ( n , F ) is called globally irreducible if for each non-archimedean valuation v of F a reduction of G modulo v is absolutely irreducible. Problem 1. Let R be an arithmetic ring and M ( n , R ) the matrix ring over R . For which n there exists a finite group G ⊂ GL ( n , R ) such that the R -span of G is just M ( n , R ) ? Dmitry Malinin On the arithmetic of integral representations

Globally irreducible representations over arithmetic rings Problem 2. Let G be a finite irreducible linear group over C , and let R be the ring spanned by the traces of the elements of G . Under what conditions � G � R ∼ = M ( n , R ) ? Problem 3 . Determine globally irreducible finite subgroups of GL ( n , C ) . The case of prime n is of particular importance for Problem 1. Lemma . Let G ⊂ GL ( k , R ) and � G � R = M ( k , R ) . Then there exists H m ⊂ GL ( km , R ) such that � H m � R = M ( km , R ) . Dmitry Malinin On the arithmetic of integral representations

Globally irreducible representations over arithmetic rings F . Van Oystaeyen, A.E. Zalesskii, Finite groups over arithmetic rings and globally irreducible representations, J. Algebra, vol. 215, p. 418-436 Theorem 1 . M ( n , Z ) is a Schur ring if and only if n is a multiple of 8 or n = 1 . If n is a multiple of 8 then M ( n , R ) is a Schur ring for every R . Observe that it is sufficient to examine maximal finite subgroups of GL ( n , R ) in order to check whether M ( n , R ) is a Schur ring. Theorem 2. Let G ⊂ GL ( n , R ) . Then � G � R = M ( n , R ) if and only if G is globally irreducible. One may expect that the existence of a globally irreducible subgroup G ⊂ GL ( n , K ) should imply that M ( n , R ) is a Schur ring. However, G is not always conjugate to a subgroup of GL ( n , R ) . Examples: Cliff, Ritter, Weiss, Feit, Serre. Dmitry Malinin On the arithmetic of integral representations