On Numbers of Tuples of Nilpotent Matrices over Finite Fields under Simultaneous Conjugation
aa r X i v : . [ m a t h . R T ] F e b On Numbers of Tuples of Nilpotent Matrices overFinite Fields under Simultaneous Conjugation
Jiuzhao Hua
Abstract
The problem of classifying tuples of nilpotent matrices over a fieldunder simultaneous conjugation is considered “hopeless”. However, fora fixed matrix order over a finite field, the number of concerned orbitsis always finite. This paper gives a formula to calculate the numberof absolutely indecomposable orbits using the same methodology fromHua [3]; those orbits are non-splittable over field extensions. As a conse-quence, those numbers are always polynomials with rational coefficients.A “generalized Weyl-Kac Denominator Identity” is derived which linksthe functions counting nilpotent matrices commuting with minimal build-ing blocks of the conjugacy classes of the General Linear Group and thecoefficients of those polynomials. It is conjectured that the coefficients ofthose polynomials are always non-negative integers.
Let q be a prime power and F q be the finite field with q elements. For anypositive integer n , let M n ( F q ) be the matrix algebra which consists of all n × n matrices over F q , GL ( n, F q ) ⊂ M n ( F q ) be the General Linear Group consistingof all invertible ones and N n ( F q ) ⊂ M n ( F q ) be the subset of all nilpotent ones.Let g be a fixed positive integer and M n ( F q ) g be the set of all g -tuples of n × n matrices over F q and N n ( F q ) g be the set of all g -tuples of nilpotent ones,i.e., M n ( F q ) g = { ( M , M , . . . , M g ) | M i ∈ M n ( F q ) , ≤ i ≤ g } , N n ( F q ) g = { ( M , M , . . . , M g ) | M i ∈ N n ( F q ) , ≤ i ≤ g } .GL ( n, F q ) acts on M n ( F q ) g by simultaneous conjugation, i.e., GL ( n, F q ) × M n ( F q ) g
7→ M n ( F q ) g ( T, ( M , M , . . . , M g )) ( T − M T, T − M T, . . . , T − M g T ) . It is obvious that N n ( F q ) g is closed under the action of GL ( n, F q ). Every g -tupleof matrices ( M , M , . . . , M g ) ∈ M n ( F q ) g gives rise to a representation of the1ree algebra F q h x , x , . . . , x g i by the following mapping: F q h x , x , . . . , x g i 7→ M n ( F q ) x i M i ( i = 1 , , · · · , g ) . Conversely, any finite dimensional representation of F q h x , x , . . . , x g i is deter-mined by a g -tuple from M n ( F q ) g . It is obvious that two g -tuples are in thesame orbit if and only if their corresponding representations are isomorphic. Definition 1.1.
An orbit of M n ( F q ) g /GL ( n, F q ) is said to be indecomposableif its corresponding representation of the free algebra F q h x , x , . . . , x g i is inde-composable; it is absolutely indecomposable if its corresponding representationis absolutely indecomposable. Thus, an orbit GL ( n, F q ) · ( M , M , . . . , M g ) is absolutely indecomposable ifthere does not exist an invertible matrix T over F q , the algebraic closure of F q ,such that( T − M T, T − M T, . . . , T − M g T ) = (cid:18)(cid:20) A B (cid:21) , (cid:20) A B (cid:21) , · · · , (cid:20) A g B g (cid:21)(cid:19) , where A i , B i for 1 ≤ i ≤ g are square matrices over F q .Let M g ( n, q ) ( I g ( n, q ), A g ( n, q )) be the number of orbits (indecomposableorbits, absolutely indecomposable orbits respectively) of g -tuples of nilpotent n × n matrices over F q under simultaneous conjugation. It will be shown inlater sections that M g ( n, q ) , I g ( n, q ) and A g ( n, q ) are all polynomials in q withrational coefficients. The case for general g -tuples has been studied in Hua[3]. It turns out that A g ( n, q )’s are of significant importance because of theirdeep connections with Geometric Invariant Theory, Quantum Group Theoryand Representation Theory of Kac-Moody Algebras (Kac [4], Ringel [5] andHausel [2]). GL ( n, F q ) Let N be the set of all positive integers, P be the set of partitions of all positiveintegers, i.e., P = { ( λ , λ , . . . , λ k ) | k ∈ N , λ i ∈ N , λ i ≥ λ i +1 ≥ , ≤ i ≤ k } . The unique partition of 0 is (0). Let Φ the set of monic irreducible polynomialsin F q [ x ] with x excluded. A partition valued function on Φ is a function δ : Φ (0) } . δ has finite support if δ ( f ) = (0) except for finitely many f in Φ.Let f ( x ) = a + a x + a x + · · · + a n − x n − + x n ∈ F q [ x ] and c ( f ) be its2 ompanion matrix , i.e., c ( f ) = . . .
00 0 1 . . . . . . − a − a − a . . . − a n − . For any m ∈ N \{ } , let J m ( f ) be the Jordan block matrix of order m with c ( f )on the main diagonal, i.e., J m ( f ) = c ( f ) I . . . c ( f ) I . . . . . . I . . . c ( f ) m × m , where I is the identity matrix of order deg( f ). For λ = ( λ , λ , . . . , λ k ) ∈ P , let J λ ( f ) be the direct sum of J λ i ( f ), i.e., J λ ( f ) = J λ ( f ) ⊕ J λ ( f ) ⊕ · · · ⊕ J λ k ( f ) , which stands for J λ ( f ) 0 . . . J λ ( f ) . . . . . . J λ k ( f ) . Rational Canonical Form Theorem implies that, for any matrix M ∈ GL ( n, F q ), there exists a unique partition valued function δ on Φ with finitesupport such that P f ∈ Φ deg f · | δ ( f ) | = n and M is conjugate to M f ∈ Φ ,δ ( f ) =(0) J δ ( f ) ( f ) . For this reason, J λ ( f ) where λ ∈ P and f ∈ Φ are called minimal building blocks of conjugacy classes of GL ( n, F q ). Rational Canonical Form for non-invertiblematrices does exist as long as Φ admits x as its member. Any partition λ ∈ P can be written in its “ exponential form ” (1 n n n · · · ),which means there are exactly n i parts in λ equal to i for all i ≥
1. The weight of λ , denoted by | λ | , is P i ≥ in i , and the length of λ , denoted by l ( λ ), is P i ≥ n i .Let ϕ r ( q ) = (1 − q )(1 − q ) · · · (1 − q r ) for r ∈ N and ϕ ( q ) = 1. Furthermore,define b λ ( q ) = Q i ≥ ϕ n i ( q ). 3 efinition 3.1. For any matrix of order m × n , the arm length of index ( i, j ) is one plus the number of minimal moves from ( i, j ) to (1 , n ) , where diagonalmoves are not permitted. Thus the arm length distribution is as follows: n n − . . . n + 1 n . . . n + 2 n + 1 . . . ... ... ... ... ... ... m + n m + n − . . . m + 2 m + 1 m m × n . The arm rank of a matrix M = [ a ij ] of order m × n , denoted by ar ( M ) , is thelargest arm length of indexes of non-zero elements of M , i.e., ar ( M ) = max { arm length of ( i, j ) | a ij = 0 where ≤ i ≤ m, ≤ j ≤ n } . Definition 3.2.
A matrix M = [ a ij ] of order m × n is of type-U if it satisfiesthe following conditions: • a ij = a st if ( i, j ) and ( s, t ) have the same arm length, • the arm rank of M is at most min( m, n ) . Thus a type-U matrix has either the following form when m ≥ n : a a . . . a n − a n a . . . a n − a n − ... ... . . . ... ...0 0 . . . a a . . . a . . . . . . m × n , or the following form when m ≤ n : . . . a a . . . a m − a m . . . a . . . a m − a m − ... ... ... ... ... . . . ... ...0 . . . . . . a a . . . . . . a m × n . Theorem 3.1 (Turnbull & Aitken [6]) . Let λ = ( λ , λ , . . . , λ k ) be a partitionwith λ ≥ λ ≥ · · · ≥ λ k ≥ and f ( x ) = x − a with a ∈ F q , then any matrixover F q that commutes with J λ ( f ) can be written as a k × k block matrix in thefollowing form: U U . . . U k U U . . . U k ... ... . . . ... U k U k . . . U kk , here submatrix U ij is a type-U matrix over F q of order λ i × λ j for all ( i, j ) where ≤ i, j ≤ k . As an example, let λ = (3 , ,
2) and f ( x ) = x − t ∈ F q [ x ], E a generic matrixthat commutes with J λ ( f ), then J λ ( f ) = t t t t t t
10 0 0 0 0 0 t , E = a b c l m p q a b l p a r s d e h i r d h u v j k f g u j f . Theorem 3.2 (Fine & Herstein [1]) . For any positive integer n , the number ofnilpotent n × n matrices over F q is equal to q n − n . For a partition λ = ( λ , λ , λ , . . . ), let λ ′ = ( λ ′ , λ ′ , λ ′ , . . . ) be its conjugatepartition , which means that λ ′ i is the number of parts in λ that are greater thanor equal to i for all i ≥ Definition 3.3.
Let λ, µ be two partitions and λ ′ = ( λ ′ , λ ′ , λ ′ , . . . ) , µ ′ =( µ ′ , µ ′ , µ ′ , . . . ) be their conjugate partitions. The“inner product” of λ and µ isdefined as follows: h λ, µ i = X i ≥ λ ′ i µ ′ i . Lemma 3.1 (Hua [3]) . Let λ = (1 m m m · · · ) and µ = (1 n n n · · · ) betwo partitions in their “exponential form”, then there holds: h λ, µ i = X i ≥ X j ≥ min ( i, j ) m i n j . Theorem 3.3.
For any partition λ ∈ P and f ( x ) = x − a ∈ F q [ x ] , the numberof nilpotent matrices over F q that commute with J λ ( f ) is q h λ,λ i− l ( λ ) . Proof.
Suppose that λ = ( λ , λ , . . . , λ k ) with λ ≥ λ ≥ · · · ≥ λ k ≥ E be the set of all matrices over F q that commute with J λ ( f ), i.e., E = { M ∈ M v ( F q ) | M J λ ( f ) = J λ ( f ) M, v = | λ |} . E is indeed the endomorphism algebra of the representation of F q [ x ] induced by J λ ( f ). Theorem 3.1 implies that E = U U . . . U k U U . . . U k ... ... . . . ... U k U k . . . U kk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) U ij is a type-U matrix over F q ,U ij is of order λ i × λ j for ≤ i, j ≤ k . Let (1 n n n · · · ) be the “exponential form” for λ . Thus there are n i partsequal to i . E has finite dimension over F q , the dimension that is contributed bythe submatrices of order i × j for all paris ( i, j ) is:5 min( i, j ) n i n j if i = j , • in i if i = j .Thus the dimension of E is: X i ≥ X j ≥ ,j = i min( i, j ) n i n j + X i ≥ in i = X i ≥ X j ≥ min( i, j ) n i n j = h λ, λ i . It follows that the order of E is: |E| = q h λ,λ i . Let D be the subspace of E defined as follows: D = D D . . . D k D D . . . D k ... ... . . . ... D k D k . . . D kk ∈ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) D ij is a matrix of order λ i × λ j ,D ij = 0 if λ i = λ j ,D ij = aI for some a ∈ F q if λ i = λ j , and N be the subspace of E defined by: N = N N . . . N k N N . . . N k ... ... . . . ... N k N k . . . N kk ∈ E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N ij is a type-U matrix of order λ i × λ j ,ar ( N ij ) ≤ λ i − if λ i = λ j . . It is evident that E is a direct sum of D and N as a vector space. It can beverified that N is a two-sided ideal of E . Furthermore N is nilpotent, i.e., everyelement in N is nilpotent. Thus every matrix E ∈ E can be uniquely written asa sum of a matrix from D and a matrix from N , i.e., E = D + N for some D ∈ D and N ∈ N . Since N is a nilpotent two-sided ideal of E , E is nilpotent if and only if D isnilpotent.As an example, if λ = (3 , , E can be written as a sumas follows: a a a d h
00 0 0 0 d h j f
00 0 0 0 j f + b c l m p q b l p r s e i r u v k g u . D ∈ D can be viewed as a diagonal block matrix. Since there are n i parts equal to i in λ , the block corresponding to i n i is: D i = a I a I . . . a n i Ia I a I . . . a n i I ... ... . . . ... a n i I a n i I . . . a n i n i I , where I is the identity matrix of order i and all a ij ∈ F q . Thus D is nilpotentif and only if every diagonal block D i is nilpotent. Since D i is conjugate to thedirect sum of i copies of the following matrix: d i = a a . . . a n i a a . . . a n i ... ... . . . ... a n i a n i . . . a n i n i ,D i is nilpotent if and only if d i is nilpotent. Since the number of nilpotentmatrix of order n i over F q is q n i − n i by Theorem 3.2, the number of nilpotentmatrices in D is: q P i ≥ n i − n i . The dimension of N is:dim( E ) − dim( D ) = h λ, λ i − X i ≥ n i . Thus the order of N is: |N | = q h λ,λ i− P i ≥ n i . Putting it all together, the number of nilpotent matrices in E is: q h λ,λ i− P i ≥ n i · q P i ≥ n i − n i = q h λ,λ i− P i ≥ n i = q h λ,λ i− l ( λ ) . This finishes the proof.
Theorem 3.4.
For any partition λ ∈ P and any monic irreducible polynomial f ∈ F q [ x ] , the number of nilpotent matrices over F q that commute with J λ ( f ) is q d ( h λ,λ i− l ( λ )) , where d is the degree of f .Proof. Suppose that d > d = 1 has been proved in Theorem 3.3.Let c ( f ) be the companion matrix for f and h c ( f ) i be the subalgebra of M d ( F q )generated by c ( f ). Since f is the characteristic equation of c ( f ), c ( f ) satisfiesthe polynomial f , i.e., f ( c ( f )) = 0. Since f is irreducible, f is the minimalpolynomial satisfied by c ( f ). This implies that I, c ( f ) , c ( f ) , · · · , c ( f ) d − forma basis for h c ( f ) i over F q , i.e., h c ( f ) i = ( d − X i =0 a i c ( f ) i | a i ∈ F q , ≤ i ≤ d − ) . h c ( f ) i is a commutative subalgebra of M d ( F q ) and the following map isan isomorphism: F q [ x ] / ( f ( x ))
7→ h c ( f ) i x c ( f ) . Since f is irreducible, F q [ x ] / ( f ( x )) is isomorphic to the finite field F q d , andhence h c ( f ) i is a finite field with q d elements.When deg( f ) >
1, Theorem 3.1 still holds as long as all submatrices U ij take values from the finite field h c ( f ) i . All arguments in the proof of Theorem3.3 still work with F q being replaced by h c ( f ) i . Thus Theorem 3.3 implies thedesired results. Let Q be the rational number field, Q [[ X ]] be the ring of formal power seriesin X over Q , Q ( q ) be the field of rational functions in q over Q , and Q ( q )[[ X ]]be the ring of formal power series in X over Q ( q ). Let φ n ( q ) be the numberof monic irreducible polynomials with degree n in F q [ x ] with x excluded. It isknown that for any positive integer n , φ n ( q ) = 1 n X d | n µ ( d )( q nd − , where the sum runs over all divisors of n and µ is the M¨obius function. Theorem 4.1.
The following identity holds in Q [[ X ]] : ∞ X n =1 M g ( n, q ) X n = ∞ Y d =1 X λ ∈P q dg ( h λ,λ i− l ( λ )) q d h λ,λ i b λ ( q − d ) X d | λ | ! φ d ( q ) . Proof.
The method applied in Theorem 4.3 from Hua [3] still works here. In cur-rent context, the Burnside orbit counting formula is applied to N n ( F q ) g /GL ( n, F q )and the number of points fixed by J λ ( f ) is equal to q deg ( f ) g ( h λ,λ i− l ( λ )) by The-orem 3.4. Repeating the arguments there yields the desired result. Definition 4.1.
Define rational functions H g ( n, q ) for all positive integers n as follows: log X λ ∈P q g ( h λ,λ i− l ( λ )) q h λ,λ i b λ ( q − ) X | λ | ! = ∞ X n =1 n H g ( n, q ) X n , where log is the formal logarithm, i.e., log(1 + x ) = P i ≥ ( − i − x i /i . heorem 4.2. The following identity holds for all positive integer n : A g ( n, q ) = q − n X d | n µ ( d ) H g (cid:16) nd , q d (cid:17) . Proof.
This is the counterpart of Theorem 4.6 from Hua [3], same argumentsapply. H g ( n, q )’s are rational functions in q , so are A g ( n, q )’s. As A g ( n, q )’s takeinteger values for all prime powers q , A g ( n, q )’s must be polynomials in q withrational coefficients. I g ( n, q ) and M g ( n, q ) can be calculated by the followingidentities: I g ( n, q ) = X d | n d X r | d µ (cid:16) dr (cid:17) A g (cid:16) nd , q r (cid:17) , ∞ X n =1 M g ( n, q ) X n = ∞ Y n =1 (1 − X n ) − I g ( n,q ) . The first identity is the counterpart of the first identity of Theorem 4.1 from Hua[3] and the second identity is a consequence of the Krull–Schmidt Theorem fromrepresentation theory. It follows that I g ( n, q ) and M g ( n, q ) are polynomials in q with rational coefficients for all n ≥ Theorem 4.3.
Let A g ( n, q ) = P s ≥ a n,s q s where a n,s ∈ Q and a n,s = 0 forsufficiently large s . Then the following identity holds in Q ( q )[[ X ]] : X λ ∈P q g ( h λ,λ i− l ( λ )) q h λ,λ i b λ ( q − ) X | λ | = ∞ Y n =1 deg A g ( n,q ) Y s =0 ∞ Y i =0 (1 − q s + i X n ) a n,s . Proof.
This is the counterpart of Theorem 4.9 from Hua [3], same argumentsapply.In the context of representations of quivers over finite fields, a conjectureof Kac [4] states that the constant term of the polynomial counting the iso-morphism classes of absolutely indecomposable representations with a givendimension vector is the same as the root multiplicity of the dimension vectorin the corresponding Kac-Moody algebra. This conjecture was later proved byHausel [2], which confirms that Theorem 4.9 from Hua [3] is a generalized Weyl-Kac Denominator Identity. Thus Theorem 4.3 here may also be regarded as ageneralized Weyl-Kac Denominator Identity for some generalized Kac-Moodyalgebra.When g = 1, Jordan Canonical Form Theorem shows that there existsonly one indecomposable nilpotent matrix over F q of a given order up to conju-gation, where the unique conjugacy class is the Jordan matrix with eigenvalue9: . . .
00 0 1 . . . . . .
10 0 0 . . . . This implies that A ( n, q ) = 1 for all n ≥
1. Thus Theorem 4.3 amounts to thefollowing identity: 1 + X λ ∈P q − l ( λ ) b λ ( q − ) X | λ | = ∞ Y n =1 ∞ Y i =0 (1 − q i X n ) . Conjecture 4.1.
For any g ≥ and n ≥ , all coefficients of the polynomial A g ( n, q ) are non-negative integers. This conjecture is supported by the following observations generated by aPython program based on Theorem 4.2: A (1 , q ) = 1 ,A (2 , q ) = 2 q,A (3 , q ) = q + 3 q + 2 q,A (4 , q ) = q + q + q + 4 q + 2 q + 7 q + 4 q + 2 q,A (5 , q ) = q + q + q + 2 q + 2 q + 4 q + 4 q + 7 q + 8 q + 13 q +13 q + 16 q + 14 q + 7 q + 2 q,A (6 , q ) = q + q + q + 2 q + 2 q + 4 q + 3 q + 7 q + 7 q +10 q + 11 q + 19 q + 17 q + 28 q + 29 q + 39 q + 40 q +53 q + 48 q + 52 q + 40 q + 25 q + 8 q + 2 q. Acknowledgments
The author would like to thank Xueqing Chen, Bangming Deng, Jie Du andYingbo Zhang for their helpful comments and suggestions.
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An Introduction to the Theory of CanonicalMatrices , pp. 143-147, Blackie & Son, London (1948).Mathematics Enthusiast
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