Conjugacy classes of centralizers in the group of upper triangular matrices
aa r X i v : . [ m a t h . G R ] J a n January 24, 2019
CONJUGACY CLASSES OF CENTRALIZERS IN THE GROUP OF UPPERTRIANGULAR MATRICES
SUSHIL BHUNIAA
BSTRACT . Let G be a group. Two elements x, y ∈ G are said to be in the same z -class if their centralizers in G are conjugate within G . In this paper, we provethat the number of z -classes in the group of upper triangular matrices is infiniteprovided that the field is infinite and size of the matrices is at least , and finiteotherwise.
1. I
NTRODUCTION
Let G be a group. Two elements x and y in G are said to be z -equivalent, de-noted by x ∼ z y , if their centralizers in G are conjugate, i.e., Z G ( y ) = g Z G ( x ) g − for some g ∈ G , where Z G ( x ) := { y ∈ G | xy = yx } denotes centralizer of x in G . Clearly, ∼ z is an equivalence relation on G . The equivalence classes withrespect to this relation are called z -classes . It is easy to see that if two elementsof a group G are conjugate then their centralizers are conjugate thus they arealso z -equivalent. However, in general, the converse is not true. In geometry, z -classes describe the behaviour of dynamical types (see for example [9], [2],and [5]). That is, if a group G is acting on a manifold M then understanding(dynamical types of) orbits is related to understanding (conjugacy classes of)centralizers.Robert Steinberg [13] (Section 3.6 Corollary 1 to Theorem 2) proved that for areductive algebraic group G defined over an algebraically closed field, of goodcharacteristic, the number of z -classes is finite. A natural question that followed: Is the number of z -classes finite for algebraic group G defined over an arbitrary field k ? In [12], A. Singh studied z -classes for real compact groups of type G . Ravi S.Kulkarni, in [10], proved that the number of z -classes in GL n ( k ) is finite if thefield k has only finitely many field extensions of any fixed finite degree. Un-less otherwise specified, we will always assume that k is a field of char = 2 . Mathematics Subject Classification.
Key words and phrases.
Upper triangular matrices, conjugacy classes, z -classes.The author is supported by the SERB, India (No. PDF/2017/001049) and DST-RFBR jointIndo-Russian project (No. INT/RUS/RFBR/P-288). Let V be an n -dimensional vector space over a field k equipped with a non-degenerate symmetric or skew-symmetric bilinear form B . Then, in [6], it isproved that there are only finitely many z -classes in orthogonal groups O n ( k ) and symplectic groups Sp n ( k ) if k has only finitely many field extensions ofany fixed finite degree. Let k be a perfect field with a non-trivial Galois auto-morphism of order . Let V be an n -dimensional vector space over k equippedwith a non-degenerate Hermitian form H . Suppose that the fixed field k hasonly finitely many field extensions of any fixed finite degree. Then, in [1], weproved that the number of z -classes in the unitary group U n ( k ) is finite. Allthe groups mentioned so far are special types of reductive algebraic groups. Anatural problem to study would be to consider z -classes in non-reductive ones.In particular, one may consider the following problem: Problem 1.1.
Is the number of z -classes finite for a solvable algebraic group?Let G be a connected solvable linear algebraic group over an algebraicallyclosed field k , then by Lie-Kolchin theorem (see Theorem 17.6 [7]) G is a sub-group of the group of upper triangular matrices in GL n ( k ) for some n .Let B n ( k ) denote the group of upper triangular matrices in GL n ( k ) . In thispaper, we solve the Problem 1.1 for this special classes of groups. In a sequel,we will do this for general nilpotent and solvable groups. The main result ofthis paper is the following theorem, which solves Problem 1.1 for the group ofupper triangular matrices: Theorem 1.2. (1)
For ≤ n ≤ , the number of z -classes in B n ( k ) is finite. (2) For n ≥ , the number of z -classes in B n ( k ) is infinite. In Section 2, we explore the semisimple z -classes for the group of upper trian-gular matrices. In Section 3, we study unipotent conjugacy classes and unipo-tent z -classes in B n ( k ) . In Section 4, we prove our main theorem of this paper.Throughout the paper, we assume that k is an infinite field of char k = 2 . Herewe include an appendix, which contains an explicit computation of unipotentconjugacy classes and their representatives in the group of upper triangular ma-trices B n ( k ) for n = 2 , , , using Belitskii’s algorithm. ENTRALIZER CLASSES IN THE GROUP OF UPPER TRIANGULAR MATRICES 3
2. S
EMISIMPLE z - CLASSES IN B n ( k ) Let n be a positive integer with a partition λ = (1 k k . . . n k n ) , denoted by λ ⊢ n , i.e., n = P i ik i . Proposition 2.1.
The number of semisimple z -classes in B n ( k ) is X (1 k k ...n kn ) ⊢ n n ! Q nj =1 ( j !) k j ( k j !) . So, in particular, the number of semisimple z -classes in B n ( k ) is finite.Proof. Semisimple elements in B n ( k ) are nothing but diagonals up to conju-gacy (for details see the first-page second paragraph in [11]). So the numberof semisimple z -classes in B n ( k ) is equal to the number of z -classes of diagonalsin B n ( k ) . Now we give a combinatorial argument to count the z -classes of di-agonals in B n ( k ) , which is as follows: Let λ = (1 k k . . . n k n ) be a partition of n .Let us consider the following multiset M := { a , a , . . . , a k | {z } k ; a , a , . . . , a k , a k | {z } k ; . . . , a i , . . . , a i , . . . , a ik i , . . . , a ik i | {z } ik i ; . . . } . So the number of ways to order the above multiset is n !(1!) k (2!) k · · · ( n !) k n . Now look at the action of S k × S k × · · · × S k n on the ordered tuples via ( σ × σ × · · · × σ n )( · · · a ij · · · ) = ( · · · a iσ i ( j ) · · · ) , where σ i ∈ S k i and S k i ’s are the symmetric groups on k i symbols. Thereforethe size of each orbit under the above action is Q nj =1 ( k j !) as the stabilizer is theidentity group. So the number of orbits is n ! Q nj =1 ( j !) k j Q nj =1 ( k j !) . Hence the result. (cid:3)
A numerical example should make the above argument transparent.
Example 2.2.
Let n = 5 , then the number of partitions of n is equal to and aregiven by (5 ) , (1 ) , (2 ) , (1 ) , (1 ) , (1 ) , (1 ) . Now SUSHIL BHUNIA (1) For λ = (5 ) ⊢ , the number of semisimple z - classes is = 1 andrepresentative is the following: diag ( α, α, α, α, α ) . (2) For λ = (1 ) ⊢ , the number of semisimple z - classes is = 5 andrepresentatives are given by the following: diag ( α, α, α, α, β ); diag ( α, α, α, β, α );diag ( α, α, β, α, α ); diag ( α, β, α, α, α );diag ( β, α, α, α, α ) . (3) For λ = (2 ) ⊢ , the number of semisimple z - classes is = 10 andrepresentatives are given by the following: diag ( α, α, α, β, β ); diag ( α, α, β, α, β );diag ( α, β, α, α, β ); diag ( β, α, α, α, β );diag ( β, α, α, β, α ); diag ( β, α, β, α, α );diag ( β, β, α, α, α ); diag ( α, α, β, β, α );diag ( α, β, β, α, α ); diag ( α, β, α, β, α ) . (4) For λ = (1 ) ⊢ , the number of semisimple z -classes is = 10 and representatives are given by the following: diag ( α, α, α, β, γ ); diag ( α, α, β, α, γ );diag ( α, β, α, α, γ ); diag ( β, α, α, α, γ );diag ( β, α, α, γ, α ); diag ( β, α, γ, α, α );diag ( α, β, γ, α, α ); diag ( α, α, β, γ, α );diag ( α, β, α, γ, α ); diag ( β, γ, α, α, α ) . (5) For λ = (1 ) ⊢ , the number of semisimple z -classes is = 15 andrepresentatives are given as follows: diag ( α, α, β, β, γ ); diag ( α, β, α, β, γ ); diag ( β, α, α, β, γ );diag ( β, α, β, γ, α ); diag ( α, β, β, γ, α ); diag ( γ, α, α, β, β );diag ( α, α, β, γ, β ); diag ( α, α, γ, β, β ); diag ( α, β, γ, α, β );diag ( α, β, γ, β, α ); diag ( α, γ, α, β, β ); diag ( α, γ, β, α, β );diag ( α, γ, β, β, α ); diag ( γ, α, β, α, β ); diag ( γ, β, α, α, β ) . ENTRALIZER CLASSES IN THE GROUP OF UPPER TRIANGULAR MATRICES 5 (6) For λ = (1 ) ⊢ , the number of semisimple z -classes is = 10 and representatives are given by the following: diag ( α, α, β, γ, δ ); diag ( β, γ, δ, α, α );diag ( β, γ, α, δ, α ); diag ( β, γ, α, α, δ );diag ( β, α, γ, α, δ ); diag ( β, α, α, γ, δ );diag ( α, β, α, γ, δ ); diag ( α, β, γ, α, δ );diag ( α, β, γ, δ, α ); diag ( β, α, γ, δ, α ) . (7) For λ = (1 ) ⊢ , the number of semisimple z -classes is = 1 and therepresentative is given as follows: diag ( α, β, γ, δ, η ) . Therefore the total number of semisimple z -classes in B ( k ) is .3. U NIPOTENT z - CLASSES IN B n ( k ) Let α ∈ k , define x α = α . Let u α = I + x α ∈ B ( k ) be a unipotent element. The following result is alreadyknown. We record this result as we are going to use it. Proposition 3.1. (1)
For ≤ n ≤ , the number of unipotent conjugacy classes in B n ( k ) is finite. Inparticular, the numbers are , , , for n = 2 , , , respectively. (2) For n ≥ , the number of unipotent conjugacy classes in B n ( k ) is infinite.Proof. (1) Use Belitskii’s algorithm, for details see [3], [8] and [14]. Also, seethe appendix for explicit calculations of the number of unipotent con-jugacy classes. We also give representatives of the unipotent conjugacyclasses in B n ( k ) for ≤ n ≤ . SUSHIL BHUNIA (2) The proof was originally given by M. Roitman in [11] for n = 12 . Thenlatter Djokovic and Malzan, in [4], proved this for n = 6 and in fact, itis the minimum value for which this happens to be true by part (1). Forcompleteness, we will give this prove again. It is enough to prove this for n = 6 . Let u α and u β are conjugate in B ( k ) , i.e., P u α P − = u β for some P ∈ B ( k ) . Then P x α = x β P . Let P = ( p ij ) , then we get the following: p = p = 0 p = p p = p p = p αp = p + βp . Therefore from the above equation, we get α = β . So the number ofunipotent conjugacy classes in B ( k ) is infinite as k is an infinite field.Hence it is true for B n ( k ) provided n ≥ . (cid:3) Corollary 3.2.
For ≤ n ≤ , the number of unipotent z -classes in B n ( k ) is finite.Proof. If two elements are conjugate, then they are also z -conjugate. So this fol-lows from the first part of Proposition 3.1. (cid:3) Now the centralizer of u α , Z B ( k ) ( u α ) is the following: a b b b b b a b − αb b + b − αb b − αb a b b b a α + 1) b − b − b a b + b − αb a | a ∈ k × , b i ∈ k . Lemma 3.3.
For n ≥ , the number of unipotent z -classes in B n ( k ) is infinite.Proof. It is enough to prove this lemma for n = 6 . Now assume that n = 6 .Suppose that u α and u β are z -conjugate, then P Z B ( k ) ( u α ) P − = Z B ( k ) ( u β ) forsome P ∈ B ( k ) . Claim : α = β .Now P AP − ∈ Z B ( k ) ( u β ) for all A ∈ Z B ( k ) ( u α ) . So P AP − = A ′ for some A ′ ∈ Z B ( k ) ( u β ) . Observe that two upper triangular matrices are conjugate via ENTRALIZER CLASSES IN THE GROUP OF UPPER TRIANGULAR MATRICES 7 a upper triangular matrix implies that they have the same diagonal entries. Let P = ( p ij ) ∈ B ( k ) and A, A ′ have the form described as above. Then we get a ′ = a (3.1) b ′ = b p p − (3.2) b ′ = ( b p − b p p p − ) p − (3.3) b ′ = b p p − (3.4) b ′ = ( b p − b p p p − ) p − . (3.5) ( b − αb ) p + b p = ( b ′ − βb ′ ) p . (3.6) (( α + 1) b − b − b ) p + ( b + b − αb ) p = (( β + 1) b ′ − b ′ − b ′ ) p . (3.7)From Equation (3.6), we get, p = 0 and αp = βp .(If b = b = 0 and b = 0 , then b ′ = 0 and b ′ = − b p p p − p − . So − b p p p − p − p = 0 , hence p = 0 , since b = 0 and p ii = 0 . And if b = b = 0 and b = 0 , then b ′ = 0 = b ′ and b ′ = b p p − . So − αb p + b p = − βb p p p − . Now, since p = 0 , we get αb p = βb a , which implies αp = βp , as b = 0 ).From Equation (3.7), we get, p = p .(If b = b = b = 0 and b = 0 , then b ′ = b ′ = b ′ = 0 and b ′ = b p p − . Sofrom Equation (3.6) we get p − p = p p p − , since b = 0 . Again if b = b = b = 0 and b = 0 , then b ′ = b ′ = 0 and b ′ = b p p − ; b ′ = − b p p p − p − = 0 , as p = 0 . So again from Equation (3.6)we get, p − p = p p p − , since b = 0 . Therefore from the above two we get p = p ). Hence α = β .Therefore the result is true for n = 6 . Now for n > . Let U α := u α I n − ! , U β := u β I n − ! ∈ B n ( k ) SUSHIL BHUNIA be two unipotent elements which are z -conjugate in B n ( k ) . Then Q Z B n ( k ) ( U α ) Q − = Z B n ( k ) ( U β ) for some Q ∈ B n ( k ) . Therefore QCQ − = C ′ for some C ∈ Z B n ( k ) ( U α ) and C ′ ∈ Z B n ( k ) ( U β ) . Now write Q, C and C ′ in block form, we get P ∗ ∗ ! A ∗ ∗ ! = A ′ ∗ ∗ ! P ∗ ∗ ! for some P, A, A ′ ∈ B ( k ) . Hence P A = A ′ P , which reduces to the case of n = 6 .Therefore the number of unipotent z -classes in B n ( k ) ( n ≥ ) is infinite. (cid:3) Corollary 3.4.
The unipotent z -classes for B ( k ) is parametrized by elements of thefield k .Proof. Let u be a unipotent element of B ( k ) . Then using Belitskii’s algorithm(see appendix and [8] for details) we get bub − = u α for some b ∈ B ( k ) and forsome α ∈ k , where u α is defined at the beginning of this section. Again fromLemma 3.3 we have u α ∼ z u β if and only if α = β , where α, β ∈ k . Thereforeunipotent z -classes in B ( k ) are completely determined by the elements of k viathe map α u α . (cid:3) z - CLASSES IN B n ( k ) Lemma 4.1.
Let g ∈ G and g = g s g u be the Jordan decomposition of g , and α ∈ G .Then we have α Z Z G ( g s ) ( g u ) α − = Z α Z G ( g s ) α − ( αg u α − ) . Proof.
Let x ∈ Z Z G ( g s ) ( g u ) then xg s = g s x and xg u = g u x . Therefore αxg u α − = αg u xα − implies that ( αxα − )( αg u α − ) = ( αg u α − )( αxα − ) . Therefore αxα − ∈Z α Z G ( g s ) α − ( αg u α − ) .On the other hand let y ∈ Z α Z G ( g s ) α − ( αg u α − ) , then ( α − yα ) g s = g s ( α − yα ) and y ( αg u α − ) = ( αg u α − ) y . Now the last equation is same as ( α − yα ) g u = g u ( α − yα ) . So Z α Z G ( g s ) α − ( αg u α − ) ⊆ α Z Z G ( g s ) ( g u ) α − . Hence the result. (cid:3) Remark 4.2.
Let us assume that the number of semisimple z -classes in G is n andrepresentatives are given by s , s , . . . , s n . Let g ∈ G then g = g s g u is the Jordandecomposition of g . By the above assumption, g s will be z -conjugate to s i for ENTRALIZER CLASSES IN THE GROUP OF UPPER TRIANGULAR MATRICES 9 some i = 1 , , . . . , n . Without loss of generality, say g s ∼ z s , i.e., α Z G ( g s ) α − = Z G ( s ) for some α ∈ G . Then α Z G ( g ) α − = α Z Z G ( g s ) ( g u ) α − = Z Z G ( s ) ( αg u α − ) . The first equality follows from the uniqueness of the Jordan decomposition, i.e., Z G ( g ) = Z G ( g s ) ∩ Z G ( g u ) = Z Z G ( g s ) ( g u ) , and the second equality follows from Lemma 4.1. Now if the number of unipo-tent z -classes in Z G ( s i ) is finite for all i = 1 , , . . . , n . Then the number of z -classes in G is finite. So the upshot is the following:If we know that the number of semisimple z -classes in G is finite, and thenumber of unipotent z -classes in centralizer of semisimple elements is finite,then the number of z -classes in G is finite.4.1. Proof of the Theorem 1.2: (1) The number of semisimple z -classes in B n ( k ) is finite follows from Propo-sition 2.1. The number of unipotent z -classes in B n ( k ) is finite for ≤ n ≤ follows from Corollary 3.2. So the number of z -classes in B n ( k ) , for ≤ n ≤ , is finite follows from Lemma 4.1 (see also Remark 4.2).(2) The number of unipotent z -classes in B n ( k ) is infinite for n ≥ followsfrom Lemma 3.3. Therefore the number of z -classes in B n ( k ) is infiniteprovided n ≥ . 5. A PPENDIX
Two matrices
A, B ∈ B n ( k ) are said to be conjugate if B = P AP − for some P ∈ B n ( k ) . Here we are following [8]. Belitskii’s Algorithm for B n ( k ) :Let A = ( a ij ) ∈ B n ( k ) . Elements of the matrix A are ordered by a nn ; a n − n − , a n − n ; . . . ; a , a , . . . , a n , i.e., a sequence from bottom to top and in each row from left to right. AIM : The aim of this algorithm is to simplify the first entry in the above se-quence, then the second entry and so on. By “simplifying” we mean replacingthe entry by or (conjugating the matrix A by upper triangular matrices) ifpossible. If not, then we continue with the next entry in the above sequence. Ateach step, we take care not to disturb any of the reductions obtained so far. Let e ij ( α ) be an elementary matrix, with ( i, j ) th element equal to α and ev-erywhere else. Two matrices A and B are conjugate if and only if one reduces tothe other by a sequence of the following two elementary transformations:(1) Multiply row i by α = 0 , then multiply column i by α − ; the elementarytransformations can be obtained as A P AP − , P = I + e ii ( α − .(2) For i < j , multiply I + e ij ( α ) from the left; then multiply I + e ij ( − α ) from the right; the elementary transformations can be obtained as A P AP − , P = I + e ij ( α ) . Algorithm for B n ( k ) (2 ≤ n ≤ :Step 1: Let a pq be the first unreduced entry of A . If the column of a pq containsan entry a iq = 1 located under a pq (i.e., i > p ) and a iq is the first nonzero entry inthe row, then a pq = 0 by transformation of the type (2) with P = I + e pi ( − a pq ) . Step 2:
Suppose that the column of a pq does not contain such an entry a iq . If a pq is the first nonzero entry of that row, then a pq = 1 by transformation of the type(1) with P = I + e pp ( a − pq − . Step 3:
Suppose a pr = 1 is the first nonzero entry in the row of a pq , then a pq = 0 by the transformation of the type (2) with P = I + e rq ( a pq ) . But this might disturbthe row a r ∗ , which was reduced before. However, this does not happen if therow a q ∗ is zero. Step 4: If a ri = 1 = a qj are the first nonzero entries of corresponding rows and i < j , then the above transformation by P = I + e rq ( a pq ) disturbs the row of a ri ,which can be restored by the transformation of the type (2) with P = I + e ij ( a pq ) (this transformation does not disturb the reduced entries since the matrix is × this need not be true for × matrices). In this case, we get a pq = 0 . Step 5: If a qj is the first nonzero entry of the row and a ri = 0 for all i < j , then a pq = 1 by transformation of the type (1) with P = I + e qq ( a pq − . But thistransformation disturbs row of a qj by changing a qj = 1 into a qj := a pq . This canbe restored by a transformation of type (1) with P = I + e jj ( a pq − . The lattertransformation does not disturb already reduced entries if j th row is zero. If j th row is not zero, then we have j = 4 since the dimension is and the element a = 1 was changed to a := a pq . This can be restored by a transformation ofthe type (1) with P = I + e ( a pq − .Here we use Belitskii’s algorithm (for details see [8]) for unipotent elements.Number of unipotent conjugacy classes in B ( k ) is and representatives are ENTRALIZER CLASSES IN THE GROUP OF UPPER TRIANGULAR MATRICES 11 the following: ; ; ENTRALIZER CLASSES IN THE GROUP OF UPPER TRIANGULAR MATRICES 13 ; ; . Therefore we have obtained also all unipotent conjugacy classes for B n ( k ) for n = 2 , , . In particular, all we have to do is to look for the bottom right × , × and × corners of the above × case.For n = 2 , representatives are the following: ! ; ! .For n = 3 , representatives are the following: ; ; . For n = 4 , representatives are the following: ; ; ; . Acknowledgement:
The author would like to acknowledge Dr. AnupamSingh and Dr. Rohit Joshi of IISER Pune for many helpful discussions. Theauthor thanks Dr. Pranab Sardar and Dr. Krishnendu Gongopadhyay of IISERMohali for encouragement. R
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