Orbits of the Centralizer of a Linear Operator
aa r X i v : . [ m a t h . D S ] O c t Orbits of the Centralizer of a Linear Operator
Paul Best, Marco Gualtieri, Patrick Hayden
Abstract.
We describe the orbit structure for the action of the centralizergroup C ( T ) of a linear operator T on a finite-dimensional complex vector space V . The main application is to the classification of solutions to a system offirst-order ODEs with constant coefficients. We completely describe the latticestructure on the set of orbits and provide a generating function for the numberof orbits in each dimension. Mathematics Subject Classification 2000:
Key Words and Phrases:
Centralizer, Classification of solutions, Orbit lattice.
1. Introduction
Let V be a finite-dimensional complex vector space, and fix T ∈ End( V ). Considerthe system of linear ordinary differential equations with constant coefficients x ′ = T x . (1)Let S denote the set of solutions x : C → V of Equation 1. The centralizer groupof the operator T , given by C ( T ) = { U ∈ GL( V ) : U T = T U } , may also be characterized as the group of invertible operators U ∈ GL ( V ) suchthat U ◦ x ∈ S for each x ∈ S . In this way, C ( T ) acts on S , and we may considertwo solutions to be equivalent when they are in the same C ( T )-orbit.The evaluation map x x (0) defines a bijection S → V with inverse x ( t exp( tT ) x ), which intertwines the natural C ( T )-actions on S and V .Therefore, equivalence classes of solutions in S are in one-to-one correspondencewith C ( T )-orbits in V . In short, to classify solutions to Equation 1, we mustdescribe the orbit structure of V under the action of C ( T ).
2. Finitely many orbits
Consider an operator with only one Jordan block, i.e. T = λI + N , where N isnilpotent of degree n = dim V . In this case, the only operators which commute Best, Gualtieri, Hayden with T are the polynomials in T . The centralizer may be described explicitly asfollows: C ( T ) = ( n − X i =0 a i N i : a i ∈ C , a = 0 ) . As a result, the orbits of C ( T ) on V are precisely given by O i = F i − F i − , where F − = ∅ and F i = ker N i , i ≥ , defines the full flag of T -invariant subspaces associated to the nilpotent operator N . Lemma 2.1. If T has only one Jordan block, then there are exactly dim V +1 orbits O , . . . , O n of C ( T ) on V , corresponding to the full flag of invariantsubspaces F ⊂ · · · ⊂ F n = V via O i = F i − F i − . In the general case, V admits a decomposition V = ⊕ i V i such that T | V i has a single Jordan block, and the centralizer of T is more complicated, as wedescribe in Section 3. However, the product of the centralizers D ( T ) = ⊕ i C ( T | V i )is contained in C ( T ). There are only finitely many orbits of D ( T ), since they areproducts of C ( T | V i )-orbits. The orbits of D ( T ), however, are refinements of theorbits of the larger group C ( T ), hence there can only be finitely many orbits ofthe centralizer group. Theorem 2.2.
There are finitely many orbits of C ( T ) in V . Let c ( T ) be the algebra of operators commuting with T . It contains thecentralizer group C ( T ) as an open dense subset, and may be identified with the Liealgebra of C ( T ). It follows that each orbit of C ( T ) in V is an open dense subsetof a c ( T )-invariant subspace of V . We now show, using the finiteness result above,that C ( T )-orbits are in one-to-one correspondence with c ( T )-invariant subspaces. Theorem 2.3.
Orbit closure is a bijection from the set of orbits of C ( T ) to theset of c ( T ) -invariant subspaces of V . Proof.
We show the map C ( T ) v C ( T ) v = c ( T ) v is a bijection by providingits inverse. If Y ⊂ V is c ( T )-invariant, let O Y be the complement in Y of theunion of its c ( T )-invariant proper subspaces. Theorem 2.2 ensures there are onlyfinitely many such subspaces, hence O Y is nonempty. Furthermore, O Y mustbe a union of orbits of C ( T ), but it cannot contain more than one orbit, since Y cannot contain two disjoint open dense sets. Hence the map Y
7→ O Y is therequired inverse.In view of the above bijection, we proceed to classify the C ( T ) orbits bycompletely describing the invariant subspaces for the action of the algebra c ( T )on V . est, Gualtieri, Hayden
3. The centralizer algebra of a linear operator
To identify the c ( T )-invariant subspaces of V , we need a convenient descriptionof the algebra c ( T ) itself. View the vector space V as a C [ x ]-module, where x v = T ( v ) for v ∈ V . This point of view is particularly useful for us, because ofthe following. Proposition 3.1.
A linear operator U commutes with T if and only if it is a C [ x ] -module endomorphism V → V . In other words, c ( T ) = End C [ x ] ( V ) . Let the minimal polynomial of T be Q λ p k λ λ , where p λ = ( x − λ ) andthe product is over distinct eigenvalues λ ∈ Spec( T ). The associated generalizedeigenspace decomposition is V = M λ ∈ Spec( T ) V λ , with V λ = ker( T − λ ) k λ . A priori, the endomorphism algebra decomposes as adirect sum of the components Hom C [ x ] ( V λ , V λ ′ ), but for λ = λ ′ this is the zerovector space, since a morphism φ : V λ → V λ ′ satisfies 0 = φ ( p k λ λ v ) = p k λ λ φ ( v ), and p λ is invertible on V λ ′ for λ = λ ′ . Hence we obtain the following decompositionof c ( T ): Proposition 3.2.
The centralizer algebra c ( T ) decomposes as a direct sumof centralizers of the restrictions T λ of T to the generalized eigenspaces V λ =ker( T − λ ) k λ . Consequently, orbits of the full centralizer algebra are products of orbits ofthe summands c ( T λ ), and we need only consider the case of a single eigenvalue.So, consider the case where T ∈ End( V ) has minimal polynomial ( x − λ ) k , andchoose a Jordan decomposition of V , as follows: V = V ⊕ · · · ⊕ V k , (2)where each V i = V i ⊕ · · · ⊕ V im i is a sum of m i cyclic modules with annihilator( x − λ ) i , and we take V i = 0 when m i = 0. In other words, T | V i consists of m i repeated Jordan blocks of size i . We now compute the module homomorphismsbetween individual summands of V i and V j . Proposition 3.3.
Let M i be the cyclic module C [ x ] /p i for p = ( x − λ ) , λ ∈ C .Then Hom C [ x ] ( M i , M i ′ ) = ( M i ′ for i ≥ i ′ ,p i ′ − i M i ′ for i ≤ i ′ . Proof.
Since M i , M i ′ are cyclic, φ ∈ Hom( M i , M i ′ ) is determined by [1] f for f ∈ M i ′ such that p i f = 0. For i ≥ i ′ this does not impose a condition on f ,but for i ′ > i we obtain f ∈ p i ′ − i M i ′ , as required. Best, Gualtieri, Hayden
Example 3.4.
Suppose V decomposes as V ⊕ V = C [ x ] v ⊕ C [ x ] v , whereann( v ) = ( x ) and ann( v ) = ( x ). Then ( v , x v , v , x v , x v ) is a Jordanbasis in which T has the following Jordan form: T =
01 0 01 01 0 φ ∈ c ( T ) then decomposes as φ + φ + φ + φ , where φ ij ∈ Hom( V i , V j ).By Proposition 3.3, we have φ ( v ) = ( a + bx ) v , φ ( v ) = ( c + dx + ex ) v , φ ( v ) = ( hx + kx ) v , and φ ( v ) = ( f + gx ) v , where a, b, c, d, e, f, g, h, k arearbitrary complex numbers. Writing φ in terms of the Jordan basis, we obtain: c ( T ) = a fb a g fch d ck h e d c : a, b, c, d, e, f, g, h, k ∈ C
4. Classification of c ( T ) -invariant subspaces For a single cyclic module M i = C [ x ] /p i , Lemma 2.1 shows that there are i + 1invariant subspaces for the action of c ( T ), forming a full flag F ⊂ · · · ⊂ F i = M i .We may write F l = p i − l M i . We now show that any c ( T )-invariant subspace in thesum of cyclic modules (2) decomposes into a direct sum of its projections to thecyclic summands. Theorem 4.1.
Let T ∈ End( V ) have minimal polynomial p k for p = ( x − λ ) ,and let m i be the number of Jordan blocks of size i , so that we may choose aJordan decomposition V = ⊕ ki =1 V i , where V i = V i ⊕ · · · ⊕ V im i is a sum of cyclicmodules isomorphic to C [ x ] /p i (and we set V i = { } for m i = 0 ). Then W ⊂ V is a c ( T ) -invariant subspace if and only if the following three conditions hold:1. W is a direct sum of subspaces of the form p i − l V ij .2. If p i − l V ij ⊂ W , then p i ′ − l V i ′ j ′ ⊂ W for all i ′ ≥ i and all j ′ .3. If p i − l V ij ⊂ W , then p i − l V i ′ j ′ ⊂ W for all i ′ ≤ i and all j ′ . Proof.
The projection π ij from V to each cyclic summand V ij commutes with T ; therefore π ij ∈ c ( T ). So, if W ⊂ V is c ( T )-invariant, it must contain all of itsprojections onto the cyclic summands, and we obtain W = ⊕ i,j π ij W . Moreover,each of π ij W is c ( T | V ij )-invariant and hence must coincide with some member p i − l V ij of the flag, proving part 1. W is c ( T )-invariant if and only if c ( T ) p i − l V ij ⊂ W for all summands p i − l V ij present in W . Recall that each element in c ( T ) is a sum of morphisms est, Gualtieri, Hayden φ ∈ Hom( V ij , V i ′ j ′ ). By Proposition 3.3, we see that the action mapHom( V ij , V i ′ j ′ ) ⊗ p i − l V ij −→ V i ′ j ′ is surjective onto p i ′ − i p i − l V i ′ j ′ = p i ′ − l V i ′ j ′ for i ′ ≥ i and any j ′ . It is also onto p i − l V i ′ j ′ for i ′ ≤ i and any j ′ , as required.Theorem 4.1 has a helpful interpretation as defining a poset, as we nowdescribe. First note that if p i − l V ij is contained in an invariant subspace W , then p i − l V ij ′ must also be contained for all j ′ = 1 , . . . , m i . Hence we treat the directsum ⊕ j p i − l V ij as a single subspace, which we denote by m i p i − l V i . We define apartial order on the set P = { m i p i − l V i } of these subspaces by setting A ≤ B when c ( T )( B ) contains A . By Theorem 4.1, the Hasse diagram of P is as drawnin Figure 1, in the (fictitious) situation that all multiplicities m i are nonzero. This m V (cid:127)(cid:127)(cid:127)(cid:127) ???? m V (cid:127)(cid:127)(cid:127)(cid:127) ???? m V (cid:127)(cid:127)(cid:127)(cid:127) ???? m V (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????? · · · m pV (cid:127)(cid:127)(cid:127)(cid:127) ???? m pV (cid:127)(cid:127)(cid:127)(cid:127) ???? m pV (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????? · · · m p V (cid:127)(cid:127)(cid:127)(cid:127) ???? m p V (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ????? · · · m p V (cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ???? · · ·· · · Figure 1: Poset P describing the action of c ( T ) on m i p i − l V i poset appears in the study of representations of gl n ( C ), where it is known as theGelfand-Tsetlin poset [2]. Corollary 4.2. W ⊂ V is a c ( T ) –invariant subspace if and only if it is a directsum of subspaces m i p i − l V i which form a decreasing subset in the above poset P . Of course, the linear operator T has Jordan blocks of only a finite numberof possible sizes. Hence, all but a finite number of the multiplicities m i are zero,and so the corresponding vertices in the poset P do not contribute to any c ( T )–invariant subspaces of which they are summands. As a result, the c ( T )–invariant We say I ⊂ P is decreasing if x ∈ I and y ≤ x imply that y ∈ I . Best, Gualtieri, Hayden subspaces are in bijection with the decreasing subsets of a subposet of P , definedby the vertices with nonzero multiplicities m i .Furthermore, c ( T )–invariant subspaces form a lattice, under the usual oper-ations of sum and intersection of subspaces. This lattice structure clearly coincideswith the usual lattice structure on decreasing subsets of the poset P . Summariz-ing, we obtain the following classification. Theorem 4.3.
Let T ∈ End( V ) have a single eigenvalue and Jordan blockswhose sizes define a finite subset B ⊂ N . The lattice of c ( T ) –invariant subspacesof V is isomorphic to the the lattice of decreasing subsets in P B , the subposet of P generated by the columns of length i ∈ B . Example 4.4. If T is nilpotent, with any number of Jordan blocks, but of sizes1, 3, and 5 only, then the c ( T )–invariant subspaces are in bijection with decreasingsubsets of the following subposet of the Gelfand-Tsetlin poset: • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ??????? • (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)??? ??? •• (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)??? ??? •• (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ??????? ••• Remark 4.5.
It is well-known [1] that the decreasing subsets of a poset forma distributive lattice, which is self-dual when the original poset is. As a result, wemay conclude that the lattice of c ( T )–invariant subspaces is a self-dual distributivelattice.
5. Orbit lattice
Theorem 4.3 characterizes the lattice of c ( T )–invariant subspaces, and thereforethe lattice of centralizer orbits, as the lattice of decreasing subsets of a posetconstructed entirely from the knowledge of the sizes (not the multiplicities) of theJordan blocks which occur in each generalized eigenspace. We now give a moreexplicit description of the orbit lattice, without reference to the Gelfand-Tsetlinposet. The orbit lattice is a Cartesian product of the orbit lattices in each gener-alized eigenspace V λ . We first determine the lattice Γ λ corresponding to a singlegeneralized eigenspace, using the notation from Theorem 4.3.Assume T has a single eigenvalue and let B ⊂ N be the set of sizes ofJordan blocks in the Jordan decomposition of T . For each block size i ∈ B , let C i be the corresponding column of length i in the subposet P B ⊂ P . The columnsare totally ordered ( C i , C i , . . . ) from smallest to largest, reading from left to right est, Gualtieri, Hayden P B .A decreasing subset X ⊂ P B is determined by the sequence ( X ∩ C i k )) k ∈ N , which counts the number of elements in each column. Alternatively,we may represent this information as a sequence δ X = ( δ X , δ X , . . . ) of successiveincrements, in the following way. Let δ Xk = ( X ∩ C i ) k = 1 X ∩ C i k ) − X ∩ C i k − ) k > . (3)The condition that X be a decreasing subset is easier to state in terms of thesequence δ X : for all k , 0 ≤ δ Xk ≤ ∆ k , (4)where ∆ = i and ∆ k = i k − i k − for k >
1. In other words, the intersection of X with each successive column C k must not decrease in length, and any increaseis bounded by the increment ∆ k in the total column length. Definition 5.1.
Let B ⊂ N be the set of sizes of Jordan blocks for T , for afixed eigenvalue. We define the sequence of block increments ∆ = (∆ k ) k ∈ N asfollows: ∆ = i , ∆ k = i k − i k − , for k > , where B = { i , i , . . . } , in increasing order so that i k < i k +1 for all k .We may then rephrase the condition (4) as follows. Proposition 5.2.
Equation 3 establishes a bijection between decreasing subsets X ⊂ P B and elements in [∆ ] × [∆ ] × · · · × [∆ B ] , where (∆ k ) k ∈ N is the sequence of block increments of T and [ n ] is the set { , , . . . , n } . The partial order on decreasing subsets of P B may be described as follows: X ≤ X ′ when X ∩ C k ) ≤ X ′ ∩ C k ) for all k . In terms of the correspondingsequences of increments δ X , δ X ′ , this is simply the condition δ X + · · · + δ Xk ≤ δ X ′ + · · · + δ X ′ k , for all k. This partial order defines a natural poset structure on the product Q k [∆ k ], forany sequence (∆ k ) k ∈ N of natural numbers. Definition 5.3.
Given the sequence ∆ = (∆ k ) k ∈ N of natural numbers, let[∆ k ] = { , . . . , ∆ k } and define a partial order on Γ ∆ = Q k [∆ k ] as follows: for r = ( r i ) i ∈ N and s = ( s i ) i ∈ N in Γ ∆ , r ≤ s if and only if X i ≤ k r i ≤ X i ≤ k s i , for all k ∈ N . (5) Best, Gualtieri, Hayden
We conclude with the explicit description of the full orbit lattice in termsof the posets defined above.
Theorem 5.4.
For each distinct eigenvalue λ of T ∈ End( V ) , let ∆ λ be theassociated sequence of block increments, as in Definition 5.1. Then the lattice oforbits of C ( T ) is isomorphic to the Cartesian lattice product Y λ ∈ Spec( T ) Γ ∆ λ , for Γ ∆ λ as given in Definition 5.3. Example 5.5.
Let T ∈ End( V ) be nilpotent, with Jordan blocks of sizes 1, 3,and 5 only. The sequence of block increments is then ∆ = (1 , , C ( T )–orbit lattice is given by [1] × [2] × [2], with the ordering specified by (5).The Hasse diagram of this lattice is given below. 122 iiiiiii iiiiiii JJ iiiiiii JJ iiiiiii JJ iiiiiii JJ JJ iiiiiii
012 110 iiiiii JJ JJ iiiiiii JJ iiiiiii JJ iiiiiii JJ iiiiiii iiiiiii
6. Counting orbits
By Theorem 5.4, centralizer orbits are in bijection with elements in the Cartesianproduct Y λ ∈ Spec( T ) Y k ∈ N [∆ λk ] , where the first product is over the distinct eigenvalues and the second is over thefinite number of nonzero block increments associated to a fixed eigenvalue. Thecardinality of [∆ λk ] is 1 + ∆ λk , so we obtain a simple formula for the total numberof centralizer orbits in terms of the set of Jordan block sizes in each generalizedeigenspace.In this section, we use the theory of generating functions [1] (c.f. Prop 1.4.4)to refine this count, giving the number of centralizer orbits of dimension n . Unlikethe total number of orbits, this depends on the multiplicities m i of the vertices est, Gualtieri, Hayden T has a single eigenvalue, let B = ( i , i , . . . )be the sizes of Jordan blocks in increasing order as before, and for each i k ∈ B ,let m i k be the multiplicity of the Jordan block of size i k . Let ( C i , C i , . . . ) be thecolumns of the subposet P B as before. If X ⊂ P B is a decreasing subset, then thecentralizer orbit it represents has dimension given by the sum of the X ∩ C i k ),where each term is weighted by the multiplicity m i k .As a result, the sequence of increments δ X = ( δ X , δ X , . . . ) defined by (3)can be used to compute the dimension of the orbit O X by the following formula:dim O X = m i δ X + m i ( δ X + δ X ) + · · · + m i k ( δ X + · · · + δ Xk ) + · · · . From this, we define the following generating function: let M n = P k ≥ n m i k be thetail sums of the sequence of multiplicities, and define f ( x ) = Y n ∈ N ∆ n X i =0 x iM n ! . Then the coefficient of x m in this polynomial is the number of distinct centralizerorbits of dimension m . We conclude with the generating function in the case ofmultiple eigenvalues. Theorem 6.1.
For each eigenvalue λ of T ∈ End( V ) , let (∆ λk ) k ∈ N be theassociated sequence of Jordan block increments, let ( m λi k ) k ∈ N be the sequence ofmultiplicities of Jordan blocks of size i k , as above, and let M λn = P k ≥ n m λi k be thetail sums of these multiplicities. Define the polynomial f λ ( x ) = Y k ∈ N ∆ λk X i =0 x iM λk . Then the number of orbits of the centralizer of T of dimension n is given by thecoefficient of x n in the generating function Y λ ∈ Spec( T ) f λ ( x ) . Example 6.2.
Let T be nilpotent, with Jordan blocks of sizes 1, 3, and 5 only,as in Example 5.5, and assume the multiplicity of the Jordan blocks is 1, 1, and 1respectively. The block increment sequence is then (1 , , , , , , f ( x ) = (1 + x )(1 + x + x )(1 + x + x )= 1 + x + 2 x + 2 x + 3 x + 3 x + 2 x + 2 x + x + x , yielding a total of f (1) = 18 orbits, occupying all dimensions from 0 to 9.0 Best, Gualtieri, Hayden
7. Acknowledgments
The idea to study centralizer orbits was given to us by Roger Howe during thePCMI workshop on Lie Theory in the summer of 1998, during which much of thiswork was completed. We apologize for the delay in publication. We thank RobertBryant, Chris Douglas, Mike Hill, Marcus Hum, and especially John Labute forhelpful conversations. We thank the referee for several improvements to the paper.
References [1] Stanley, R. P., “Enumerative combinatorics, Vol. 1”, Cambridge Studies inAdvanced Mathematics , Cambridge Univ. Press, Cambridge, 1997.[2] Gelfand, I. M., and M. L. Tsetlin, Finite dimensional representations of thegroup of unimodular matrices , Dokl. Akad. Nauk SSSR (N. S.) (1950),825–828.(1950),825–828.