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Commuting Pairs in the Centralizers of 2-Regular Matrices
Journal of Algebra 214, 174᎐181 Ž1999. Article ID jabr.1998.7703, available online at http:rrwww.idealibrary.com on
Commuting Pairs in the Centralizers of 2-Regular Matrices* Michael G. Neubauer and B. A. Sethuraman† Department of Mathematics, California State Uni¨ ersity Northridge, Northridge, California 91330 E-mail: [email protected], [email protected] Communicated by Susan Montgomery Received June 19, 1997
In MnŽ k ., k an algebraically closed field, we call a matrix l-regular if each eigenspace is at most l-dimensional. We prove that the variety of commuting pairs in the centralizer of a 2-regular matrix is the direct product of various affine spaces and various determinantal varieties Zl, m obtained from matrices over truncated polynomial rings. We prove that these varieties Zl, m are irreducible and apply this to the case of the k-algebra generated by three commuting matrices: we show that if one of the three matrices is 2-regular, then the algebra has dimension at most n. We also show that such an algebra is always contained in a commutative subalgebra of MnŽ k . of dimension exactly n. 䊚 1999 Academic Press
1. INTRODUCTION Recall that if k is a field, a matrix A g MnŽ k . is said to be regular Žor nonderogatory. if the minimal polynomial is equal to its characteristic polynomial, or equivalently, if V ( k n is a cyclic k w A x module, or equivalently, if each eigenspace of A is one-dimensional. Generalizing this, we will call a matrix A l-regular if each eigenspace of A is at most l-dimensional, or equivalently, if V ( k n is generated by at most l elements as a k w A x module. ŽThus, a regular matrix is now a 1-regular matrix in this terminology.. * We thank Bob Guralnick for some useful discussions and for making many clarifying suggestions, particularly in Section 5. This revised version was written when the second author was visiting the University of Southern California, and he thanks the department there for its hospitality. † The second author is supported in part by an N.S.F. grant. 174 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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We study, in this paper, the variety of commuting pairs in the centralizer of A, for A a 2-regular matrix. We show that this variety is closely related to determinantal varieties, and we prove that it is irreducible. Our goal will be the following question first raised by Gerstenhaber w3x: If A, B, and C are three commuting matrices in MnŽ k ., is dim k k w A, B, C x bounded by n? We show that the answer is yes if one of the matrices is 2-regular Žand more generally, if some two of the matrices commutes with a 2-regular matrice.. Moreover, we show that such an algebra is always contained in a commutative subalgebra of dimension exactly n, from which it follows, of course, that the centralizer of such an algebra has dimension at least n. Interestingly, we find that in contrast to the 2-generated case, it is possible for k w A, B, C x to be of dimension n but CentŽ k w A, B, C x. to be of dimension greater than n.
2. PRELIMINARIES ON 2-REGULAR MATRICES We show that the set of 2-regular Žmore generally l-regular. matrices 2 forms an open set in MnŽ k . Žviewed as An ., and we review the description of the centralizer of a 2-regular matrix. PROPOSITION 1. For any l G 1, the set of l-regular matrices in MnŽ k . 2 Ž ¨ iewed as An . is Zariski-open. Proof. We give an improvement over our own proof that is due to Guralnick: Let V ( k n. For each l-tuple p s Ž ¨ 1 , . . . , ¨ l . in V k , let UŽ p . be the set of those matrices A g MnŽ k . for which ¨ 1 , . . . , ¨ l generate V as a k w A x module. UŽ p . is an open set, since it consists of those matrices for which the n = nl matrix with columns Ai ¨ j Ž i s 0, . . . , n y 1, j s 1, . . . , l . has rank exactly n. The set of l-regular matrices is simply the union over all p of the open sets UŽ p .. Notation 2. Give a matrix C, we will let CC2 denote the variety of commuting pairs in the centralizer of C. We recall the following elementary facts, which we state without proof: LEMMA 3. Gi¨ en C g MnŽ k ., let 1 , . . . , s be its distinct eigen¨ alues. Let V ( k n decompose into V1 [ ⭈⭈⭈ [ Vs as k w C x modules, where the Vi are the elements annihilated by a suitable power of C y i I. Write Ci for C < V i . Then Ž1. End K wC x V ( End K wC x V1 = ⭈⭈⭈ = End K wC xVs . 1 s Ž2. CC2 ( CC2 = ⭈⭈⭈ = CC2 . 1 s Ž3. For any i for which Ci is 1-regular, End K wC xVi ( k w Ci x, an i algebra of dimension dim k Vi Žs t, say ., and CC2i ( At = At.
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The lemma above allows us to reduce our study of commuting pairs in the centralizer of 2-regular matrices to the following situation: C is nilpotent, with a 2-dimensional eigenspace. Thus, V decomposes as a k w C x module as V1 [ V2 , and we assume that dim k V1 s p G dim k V2 s m. Writing R for K w C x, we have R ( k w z xrz p , and V1 ( R and V2 ( Rrz m R as R-modules. The following result is classical Žsee w2, Chap. 8, Sect. 2x for instance .: LEMMA 4.
E¨ ery element of End R V is uniquely described by a matrix
ž
f 1, 1 Ž z .
z e f 1, 2 Ž z .
f 2, 1 Ž z .
f 2, 2 Ž z .
/
,
where f 1, 1 g R, f 1, 2 is in the subspace of R consisting of polynomials of degree less than m, f 2, 1 and f 2, 2 are in Rrz m R, and e s p y m. Proof (Sketch). Let u and w be generators for V1 and V2 , respectively. For any A g End R V, write Au s f 1, 1Ž z . u q f 2, 1Ž z . ¨ and Aw s f 1, 2 Ž z . u q f 2, 2 Ž z . ¨ , where f 1, 1 and f 1, 2 are uniquely determined in R and f 2, 1 and f 2,2 are uniquely determined in Rrz m R. Since z m annihilates w, f 1, 2 is constrained to be in z e R, from which the description of A follows. Note that the multiplication in End R V corresponds to an obvious multiplication of the matrices described in the lemma. Also, note that when p s m, End R V ( M2 Ž k w z xrz m . ( Ž M2 Ž k ..w z xrz m .
3. REDUCTION TO THE 2-REGULAR HOMOGENEOUS CASE We continue in the situation of Lemma 4 above and show that we may further reduce our study to the homogeneous case. ŽRecall that a matrix A is called homogeneous if it has only one eigenvalue, and all its Jordan blocks have the same size.. In the lemma below, we will abuse the notation of Lemma 4 slightly and rewrite the Ž1, 1. slot as f 1, 1Ž z . q z m f 1,X 1Ž z ., where f 1, 1 is simply the first m monomials of the Ž1, 1. slot. LEMMA 5. Gi¨ en
As
ž
f 1, 1 Ž z . q z m f 1,X 1 Ž z .
z e f 1, 2 Ž z .
f 2, 1 Ž z .
f 2, 2 Ž z .
/
,
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and Bs
ž
g 1, 1 Ž z . q z m g X1, 1 Ž z .
z e g 1, 2 Ž z .
g 2, 1 Ž z .
g 2, 2 Ž z .
/
in End R V, the matrices A and B commute if and only if the following matrices As
ž
f 1, 1 Ž z .
f 1, 2 Ž z .
f 2 , 1Ž z .
f 2, 2 Ž z .
g 1, 1 Ž z .
g 1, 2 Ž z .
g 2, 1 Ž z .
g 2, 2 Ž z .
/
and Bs
ž
/
commute in M2 Ž k w z xrz m .. Proof. This is an easy computation. COROLLARY 6. The ¨ ariety of commuting pairs in End kwC xV is just the product of Ae = Ae and the ¨ ariety of commuting pairs in End kwC xV, where e s p y m and C is nilpotent, homogeneous, and 2-regular Ž but not 1-regular ., and V is a 2 m-dimensional space isomorphic to k w C x [ k w C x as k w C x modules. Proof. This is clear.
4. THE COMMUTING PAIRS VARIETY IN THE HOMOGENEOUS CASE We now assume that C is a nilpotent homogeneous 2-regular Žbut not 1-regular. matrix and describe the equations of the variety of commuting pairs in the centralizer of C. We observe that this variety is just a determinantal variety over k w z xrz m and prove that it is irreducible. We write V ( k w C x [ k w C x as k w C x modules, with k w C x ( k w z xrz m . Letting n s dim k V, we find n s 2 m. The centralizer of C, by our earlier discussions, is isomorphic to M2 Ž k w z xrz m . ( Ž M2 Ž k ..w z xrz m . THEOREM 7. Let C be as abo¨ e. Then CC2 ( Z3, m = Am = Am , where Zl, m is the sub¨ ariety of A2 l m consisting of all l-tuples p i , q i Ž i s 0, . . . , m y 1., subject to the following equations in k l n k l : s
Ý p i n q syi s 0 Ž s s 0, . . . , m y 1. . is0
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The correspondence associates to a commuting pair Ž A, B ., with
ž
As
f 1, 1 Ž z .
f 1, 2 Ž z .
f 2, 1 Ž z .
f 2, 2 Ž z .
g 1, 1 Ž z .
g 1, 2 Ž z .
g 2, 1 Ž z .
g 2, 2 Ž z .
/
and Bs
ž
/
,
the triples
pi s
f 1, 2, i f 2, 1, i f 1, 1, i y f 2, 2, i
0
and
qi s
g 1, 2, i g 2, 1, i g 1, 1, i y g 2, 2, i
0
in Z 3, m and the point Ž f 1, 1, 0 , . . . , f 1, 1, my1 , g 1, 1, 0 , . . . , g 1, 1, my1 . in Am = Am . Ž Here, f i, j, k stands for the coefficient of z k in f i, j Ž z ... Proof. This is a simple calculation. The commuting relation AB s BA gives us the following equations in k w z xrz m : f 1, 2 g 2, 1 s g 1, 2 f 2, 1 f 1, 1 g 1 , 2 q f 1, 2 g 2, 2 s g 1, 1 f 1, 2 q g 1, 2 f 2, 2 f 2, 1 g 1, 1 q f 2, 2 g 2, 1 s g 2, 1 f 1, 1 q g 2, 2 f 2, 1 . Expanding in powers of z and collecting like terms, we have the result. Remark 8. The varieties Zl, m may be interpreted as determinantal varieties as follows: We may identify A2 l m with Ml, 2 Ž k w z xrz m . by associating to the l-tuples p i , q i Ž i s 0, . . . , m y 1. the matrix with columns 1 i my 1 i Ý my is0 p i z and Ý is0 q i z . Then, the defining equations for Zl, m are precisely the conditions that the exterior product of this matrix vanish. Our next result is key to our understanding of dim k k w A, B, C x. THEOREM 9. The ¨ arieties Zl, m defined in Theorem 7 abo¨ e are irreducible, of dimension mŽ l q 1.. Proof. It is easy to see that in the open set U where p 0 / 0, the s solution to the equations for Zl, m is given by q s s Ý is0 syi p i Ž s s 0, . . . , m y 1., where the i are arbitrary elements of k. Moreover, s s Ý is0 syi p i s Ý is0 Xsyi p i Ž s s 0, . . . , m y 1. if and only if i s Xi Ž i s 0, . . . , m y 1.. So, U is isomorphic to an open subset of Al m = Am , and is thus irreducible of dimension mŽ l q 1..
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When m s 1, Zl, 1 is just the variety of l = 2 matrices of rank at most 1, a variety that is well known to be irreducible. Now assume that we have shown that Zl, my1 is irreducible. To show that Zl, m is irreducible, it suffices to show that the set U defined above is dense in Zl, m , so it suffices to show that any point with p 0 s 0 is in the closure of U. But the equations of Zl, m show that the subvariety p 0 s 0 is just Zl, my1 = Al , which is irreducible by induction. In particular, the set where p 0 s 0 and q 0 / 0 is dense in this subvariety. It thus suffices to show that any point with p 0 s 0 and q 0 / 0 is in the closure of U. Given such a point P s Ž0, p 1 , . . . , p my1 , q 0 , . . . , q my1 . with q 0 / 0, consider the line QŽ s . s Ž sq 0 , p 1 q sq 1 , . . . , p my1 q sq my1 , q 0 , . . . , q my1 .. Using the fact that P is in Zl, m , it is easy to see that QŽ s . is in Zl, m , and of course, for s / 0, QŽ s . g U. It follows that the closure of U contains QŽ0. s P. COROLLARY 10.
For C any two regular matrix, CC2 is irreducible.
Proof. This just follows from Lemma 3, Corollary 6, and Theorems 7 and 9 above.
5. THE ALGEBRA GENERATED BY 3 COMMUTING MATRICES We will study in this section, the algebra generated by a commuting triple A, B, C, where some two of these matrices commute with a 2-regular matrix. Our approach will be via certain subsets of Cn3, the variety of commuting triples of n = n matrices. ŽWe note that for n G 32, Cn3 is known to be reducible, while for n F 4, it is known to be irreducible; see w4, 5x.. For any i, 1 F i - n, let Vi be the open subset of MnŽ k . of i-regular matrices. Let j Ž j s 1, 2, 3. represent the projection from Cn3 to the jth y1 y1 component, and let X i s y1 1 Vi , Yi s 2 Vi , and Z i s 3 Vi . Let Ui s X i l Yi l Zi . LEMMA 11. Us X is Yis Zi, where the bars denote the Zariski closures i of the respecti¨ e sets. Proof. We will prove that Us X i; the other proofs are similar. It is i sufficient to prove that X i ; U.i Given a triple Ž A, B, C . with A i-regular, the line L s Ž A, Ž1 y t . B q tA, Ž1 y t .C q tA.4 is contained in Cn3 and intersects Ui at least in the point t s 1. Since Ui is open in Cn3 and Ui l L is nonempty, Ui is dense in L. It follows that L : U.i In particular, Ž A, B, C . g U.i Remark 12. It is easy to see that Uny 1s Cn3.
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Remark 13. U1 is irreducible of dimension n2 q 2 n. This follows from the fact that X 1 is irreducible of dimension n2 q 2 n, which in turn follows from the fact that any commuting triple Ž A, B, C ., with A a 1-regular matrix, must satisfy B s f Ž A. and C s g Ž A. for uniquely determined polynomials f and g of degree at most n y 1. ŽSee w4x for instance.. We have the filtration U1: U2: ⭈⭈⭈ : Uny1s Cn3. The following is a consequence of the irreducibility of CC2 considered in the previous section. PROPOSITION 14. U2s U1, so that U2 is also irreducible of dimension n2 q 2 n. Proof. Since U1 ; U2 , it is sufficient to prove that U2; U1. By Lemma 11 above, it is sufficient to prove that X 2 : U1. Given a commuting triple Ž A, B, C ., with A a 2-regular matrix, note that there exists a point Ž D, 0. in CA2 with D a 1-regular matrix. Thus, Y1 l CA2 is nonempty Žhere, we are viewing CA2 as sitting naturally inside Cn3 .. Since CA2 is irreducible, Y1 l CA2 is all of CA2 . But Y1 l CA2; Y1s U1. Thus, Ž A, B, C . g U1. THEOREM 15. Let T be the subset of Cn3 consisting of triples Ž A, B, C . in which some two of the three matrices commute with a 2-regular matrix. For Ž A, B, C . g T, let A denote the algebra k w A, B, C x. We ha¨ e the following: Ž1.
T : U1.
Ž2. A is contained in a commutati¨ e subalgebra of MnŽ k . of dimension exactly n. Ž3.
The dimension of A is at most n.
Ž4.
The dimension of the centralizer of A is at least n.
Proof. Suppose A and B commute with a 2-regular matrix D. The line L s Ž A, B, Ž1 y t .C q tD .4 is contained in Cn3 and intersects Z2 at least in the point t s 1. Since Z2 is open in Cn3 and Z2 l L is nonempty, Z2 is dense in L. It follows that L : Z2s U2s U1, the last two equalities arising from Lemma 11 and Proposition 14. Hence Ž A, B, C . g U1. The conditions in Ž2., Ž3., and Ž4. are all polynomial conditions in the coordinates of the matrices A, B, and C. ŽThe proofs that the conditions in Ž3. and Ž4. are polynomial conditions are standard, see w8x for instance. For the condition in Ž2., see w5x.. Since these polynomial conditions hold trivially on X 1 , they hold on the closure X 1s U1. Since T ; U1, the statements immediately follow. ŽOf course, statements Ž3. and Ž4. are also direct consequences of Ž2...
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6. AN EXAMPLE We show that unlike the two-generated case Žsee w8, Theorem 1.1x., it is possible for k w A, B, C x to be of dimension n but CentŽ k w A, B, C x. to be of dimension greater than n. Let C be a nonzero nilpotent homogeneous 2-regular Žbut not 1-regular. matrix satisfying C 2 s 0. Then k w C x ( k w z xrz 2 and CentŽ C . ( Ž M2 Ž k ..w z xrz 2 . Let X 1 , X 2 , and X 3 be any three matrices in M2 Ž k . that, together with the identity matrix I2 , form a basis for M2 Ž k .. Take A s zX1 and B s zX 2 . The linear independence condition shows that k w A, B, C x is spanned by I2 , z s zI2 , zX1 , and zX 2 . On the other hand, zX 3 centralizes all four of these matrices, so dim k CentŽ k w A, B, C x. G 5. ŽIn fact, it is easy to check that dim k CentŽ k w A, B, C x. is exactly 5..
REFERENCES 1. J. Barria and P. Halmos, Vector bases for two commuting matrices, Linear and Multilinear Algebra 27 Ž1990., 147᎐157. 2. F. R. Gantmacher, ‘‘The Theory of Matrices,’’ Vol. 1, Chelsea, New York, 195. 3. M. Gerstenhaber, On dominance and varieties of commuting matrices, Ann. of Math. 73 Ž1961., 324᎐348. 4. R. Guralnick, A note on commuting pairs of matrices, Linear and Multilinear Algebra 31 Ž1992., 71᎐75. 5. R. Guralnick and B. A. Sethuraman, Commuting pairs and triples of matrices, and theorems of Motzkin᎐Taussky and Gerstenhaber, in preparation. 6. T. Laffey and S. Lazarus, Two-generated commutative matrix subalgebras, Linear Algebra Appl. 147 Ž1991., 249᎐273. 7. T. Motzkin and O. Taussky-Todd, Pairs of matrices with property L, II, Trans. Amer. Math. Soc. 80 Ž1955., 387᎐401. 8. M. Neubauer and D. Saltman, Two-generated commutative subalgebras of MnŽ F ., J. Algebra 164 Ž1994., 545᎐562. 9. A. R. Wadsworth, On commuting pairs of matrices, Linear and Multilinear Algebra 27 Ž1990., 159᎐162.
Commuting Pairs in the Centralizers of 2-Regular Matrices* Michael G. Neubauer and B. A. Sethuraman† Department of Mathematics, California State Uni¨ ersity Northridge, Northridge, California 91330 E-mail: [email protected], [email protected] Communicated by Susan Montgomery Received June 19, 1997
In MnŽ k ., k an algebraically closed field, we call a matrix l-regular if each eigenspace is at most l-dimensional. We prove that the variety of commuting pairs in the centralizer of a 2-regular matrix is the direct product of various affine spaces and various determinantal varieties Zl, m obtained from matrices over truncated polynomial rings. We prove that these varieties Zl, m are irreducible and apply this to the case of the k-algebra generated by three commuting matrices: we show that if one of the three matrices is 2-regular, then the algebra has dimension at most n. We also show that such an algebra is always contained in a commutative subalgebra of MnŽ k . of dimension exactly n. 䊚 1999 Academic Press
1. INTRODUCTION Recall that if k is a field, a matrix A g MnŽ k . is said to be regular Žor nonderogatory. if the minimal polynomial is equal to its characteristic polynomial, or equivalently, if V ( k n is a cyclic k w A x module, or equivalently, if each eigenspace of A is one-dimensional. Generalizing this, we will call a matrix A l-regular if each eigenspace of A is at most l-dimensional, or equivalently, if V ( k n is generated by at most l elements as a k w A x module. ŽThus, a regular matrix is now a 1-regular matrix in this terminology.. * We thank Bob Guralnick for some useful discussions and for making many clarifying suggestions, particularly in Section 5. This revised version was written when the second author was visiting the University of Southern California, and he thanks the department there for its hospitality. † The second author is supported in part by an N.S.F. grant. 174 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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We study, in this paper, the variety of commuting pairs in the centralizer of A, for A a 2-regular matrix. We show that this variety is closely related to determinantal varieties, and we prove that it is irreducible. Our goal will be the following question first raised by Gerstenhaber w3x: If A, B, and C are three commuting matrices in MnŽ k ., is dim k k w A, B, C x bounded by n? We show that the answer is yes if one of the matrices is 2-regular Žand more generally, if some two of the matrices commutes with a 2-regular matrice.. Moreover, we show that such an algebra is always contained in a commutative subalgebra of dimension exactly n, from which it follows, of course, that the centralizer of such an algebra has dimension at least n. Interestingly, we find that in contrast to the 2-generated case, it is possible for k w A, B, C x to be of dimension n but CentŽ k w A, B, C x. to be of dimension greater than n.
2. PRELIMINARIES ON 2-REGULAR MATRICES We show that the set of 2-regular Žmore generally l-regular. matrices 2 forms an open set in MnŽ k . Žviewed as An ., and we review the description of the centralizer of a 2-regular matrix. PROPOSITION 1. For any l G 1, the set of l-regular matrices in MnŽ k . 2 Ž ¨ iewed as An . is Zariski-open. Proof. We give an improvement over our own proof that is due to Guralnick: Let V ( k n. For each l-tuple p s Ž ¨ 1 , . . . , ¨ l . in V k , let UŽ p . be the set of those matrices A g MnŽ k . for which ¨ 1 , . . . , ¨ l generate V as a k w A x module. UŽ p . is an open set, since it consists of those matrices for which the n = nl matrix with columns Ai ¨ j Ž i s 0, . . . , n y 1, j s 1, . . . , l . has rank exactly n. The set of l-regular matrices is simply the union over all p of the open sets UŽ p .. Notation 2. Give a matrix C, we will let CC2 denote the variety of commuting pairs in the centralizer of C. We recall the following elementary facts, which we state without proof: LEMMA 3. Gi¨ en C g MnŽ k ., let 1 , . . . , s be its distinct eigen¨ alues. Let V ( k n decompose into V1 [ ⭈⭈⭈ [ Vs as k w C x modules, where the Vi are the elements annihilated by a suitable power of C y i I. Write Ci for C < V i . Then Ž1. End K wC x V ( End K wC x V1 = ⭈⭈⭈ = End K wC xVs . 1 s Ž2. CC2 ( CC2 = ⭈⭈⭈ = CC2 . 1 s Ž3. For any i for which Ci is 1-regular, End K wC xVi ( k w Ci x, an i algebra of dimension dim k Vi Žs t, say ., and CC2i ( At = At.
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The lemma above allows us to reduce our study of commuting pairs in the centralizer of 2-regular matrices to the following situation: C is nilpotent, with a 2-dimensional eigenspace. Thus, V decomposes as a k w C x module as V1 [ V2 , and we assume that dim k V1 s p G dim k V2 s m. Writing R for K w C x, we have R ( k w z xrz p , and V1 ( R and V2 ( Rrz m R as R-modules. The following result is classical Žsee w2, Chap. 8, Sect. 2x for instance .: LEMMA 4.
E¨ ery element of End R V is uniquely described by a matrix
ž
f 1, 1 Ž z .
z e f 1, 2 Ž z .
f 2, 1 Ž z .
f 2, 2 Ž z .
/
,
where f 1, 1 g R, f 1, 2 is in the subspace of R consisting of polynomials of degree less than m, f 2, 1 and f 2, 2 are in Rrz m R, and e s p y m. Proof (Sketch). Let u and w be generators for V1 and V2 , respectively. For any A g End R V, write Au s f 1, 1Ž z . u q f 2, 1Ž z . ¨ and Aw s f 1, 2 Ž z . u q f 2, 2 Ž z . ¨ , where f 1, 1 and f 1, 2 are uniquely determined in R and f 2, 1 and f 2,2 are uniquely determined in Rrz m R. Since z m annihilates w, f 1, 2 is constrained to be in z e R, from which the description of A follows. Note that the multiplication in End R V corresponds to an obvious multiplication of the matrices described in the lemma. Also, note that when p s m, End R V ( M2 Ž k w z xrz m . ( Ž M2 Ž k ..w z xrz m .
3. REDUCTION TO THE 2-REGULAR HOMOGENEOUS CASE We continue in the situation of Lemma 4 above and show that we may further reduce our study to the homogeneous case. ŽRecall that a matrix A is called homogeneous if it has only one eigenvalue, and all its Jordan blocks have the same size.. In the lemma below, we will abuse the notation of Lemma 4 slightly and rewrite the Ž1, 1. slot as f 1, 1Ž z . q z m f 1,X 1Ž z ., where f 1, 1 is simply the first m monomials of the Ž1, 1. slot. LEMMA 5. Gi¨ en
As
ž
f 1, 1 Ž z . q z m f 1,X 1 Ž z .
z e f 1, 2 Ž z .
f 2, 1 Ž z .
f 2, 2 Ž z .
/
,
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and Bs
ž
g 1, 1 Ž z . q z m g X1, 1 Ž z .
z e g 1, 2 Ž z .
g 2, 1 Ž z .
g 2, 2 Ž z .
/
in End R V, the matrices A and B commute if and only if the following matrices As
ž
f 1, 1 Ž z .
f 1, 2 Ž z .
f 2 , 1Ž z .
f 2, 2 Ž z .
g 1, 1 Ž z .
g 1, 2 Ž z .
g 2, 1 Ž z .
g 2, 2 Ž z .
/
and Bs
ž
/
commute in M2 Ž k w z xrz m .. Proof. This is an easy computation. COROLLARY 6. The ¨ ariety of commuting pairs in End kwC xV is just the product of Ae = Ae and the ¨ ariety of commuting pairs in End kwC xV, where e s p y m and C is nilpotent, homogeneous, and 2-regular Ž but not 1-regular ., and V is a 2 m-dimensional space isomorphic to k w C x [ k w C x as k w C x modules. Proof. This is clear.
4. THE COMMUTING PAIRS VARIETY IN THE HOMOGENEOUS CASE We now assume that C is a nilpotent homogeneous 2-regular Žbut not 1-regular. matrix and describe the equations of the variety of commuting pairs in the centralizer of C. We observe that this variety is just a determinantal variety over k w z xrz m and prove that it is irreducible. We write V ( k w C x [ k w C x as k w C x modules, with k w C x ( k w z xrz m . Letting n s dim k V, we find n s 2 m. The centralizer of C, by our earlier discussions, is isomorphic to M2 Ž k w z xrz m . ( Ž M2 Ž k ..w z xrz m . THEOREM 7. Let C be as abo¨ e. Then CC2 ( Z3, m = Am = Am , where Zl, m is the sub¨ ariety of A2 l m consisting of all l-tuples p i , q i Ž i s 0, . . . , m y 1., subject to the following equations in k l n k l : s
Ý p i n q syi s 0 Ž s s 0, . . . , m y 1. . is0
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The correspondence associates to a commuting pair Ž A, B ., with
ž
As
f 1, 1 Ž z .
f 1, 2 Ž z .
f 2, 1 Ž z .
f 2, 2 Ž z .
g 1, 1 Ž z .
g 1, 2 Ž z .
g 2, 1 Ž z .
g 2, 2 Ž z .
/
and Bs
ž
/
,
the triples
pi s
f 1, 2, i f 2, 1, i f 1, 1, i y f 2, 2, i
0
and
qi s
g 1, 2, i g 2, 1, i g 1, 1, i y g 2, 2, i
0
in Z 3, m and the point Ž f 1, 1, 0 , . . . , f 1, 1, my1 , g 1, 1, 0 , . . . , g 1, 1, my1 . in Am = Am . Ž Here, f i, j, k stands for the coefficient of z k in f i, j Ž z ... Proof. This is a simple calculation. The commuting relation AB s BA gives us the following equations in k w z xrz m : f 1, 2 g 2, 1 s g 1, 2 f 2, 1 f 1, 1 g 1 , 2 q f 1, 2 g 2, 2 s g 1, 1 f 1, 2 q g 1, 2 f 2, 2 f 2, 1 g 1, 1 q f 2, 2 g 2, 1 s g 2, 1 f 1, 1 q g 2, 2 f 2, 1 . Expanding in powers of z and collecting like terms, we have the result. Remark 8. The varieties Zl, m may be interpreted as determinantal varieties as follows: We may identify A2 l m with Ml, 2 Ž k w z xrz m . by associating to the l-tuples p i , q i Ž i s 0, . . . , m y 1. the matrix with columns 1 i my 1 i Ý my is0 p i z and Ý is0 q i z . Then, the defining equations for Zl, m are precisely the conditions that the exterior product of this matrix vanish. Our next result is key to our understanding of dim k k w A, B, C x. THEOREM 9. The ¨ arieties Zl, m defined in Theorem 7 abo¨ e are irreducible, of dimension mŽ l q 1.. Proof. It is easy to see that in the open set U where p 0 / 0, the s solution to the equations for Zl, m is given by q s s Ý is0 syi p i Ž s s 0, . . . , m y 1., where the i are arbitrary elements of k. Moreover, s s Ý is0 syi p i s Ý is0 Xsyi p i Ž s s 0, . . . , m y 1. if and only if i s Xi Ž i s 0, . . . , m y 1.. So, U is isomorphic to an open subset of Al m = Am , and is thus irreducible of dimension mŽ l q 1..
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When m s 1, Zl, 1 is just the variety of l = 2 matrices of rank at most 1, a variety that is well known to be irreducible. Now assume that we have shown that Zl, my1 is irreducible. To show that Zl, m is irreducible, it suffices to show that the set U defined above is dense in Zl, m , so it suffices to show that any point with p 0 s 0 is in the closure of U. But the equations of Zl, m show that the subvariety p 0 s 0 is just Zl, my1 = Al , which is irreducible by induction. In particular, the set where p 0 s 0 and q 0 / 0 is dense in this subvariety. It thus suffices to show that any point with p 0 s 0 and q 0 / 0 is in the closure of U. Given such a point P s Ž0, p 1 , . . . , p my1 , q 0 , . . . , q my1 . with q 0 / 0, consider the line QŽ s . s Ž sq 0 , p 1 q sq 1 , . . . , p my1 q sq my1 , q 0 , . . . , q my1 .. Using the fact that P is in Zl, m , it is easy to see that QŽ s . is in Zl, m , and of course, for s / 0, QŽ s . g U. It follows that the closure of U contains QŽ0. s P. COROLLARY 10.
For C any two regular matrix, CC2 is irreducible.
Proof. This just follows from Lemma 3, Corollary 6, and Theorems 7 and 9 above.
5. THE ALGEBRA GENERATED BY 3 COMMUTING MATRICES We will study in this section, the algebra generated by a commuting triple A, B, C, where some two of these matrices commute with a 2-regular matrix. Our approach will be via certain subsets of Cn3, the variety of commuting triples of n = n matrices. ŽWe note that for n G 32, Cn3 is known to be reducible, while for n F 4, it is known to be irreducible; see w4, 5x.. For any i, 1 F i - n, let Vi be the open subset of MnŽ k . of i-regular matrices. Let j Ž j s 1, 2, 3. represent the projection from Cn3 to the jth y1 y1 component, and let X i s y1 1 Vi , Yi s 2 Vi , and Z i s 3 Vi . Let Ui s X i l Yi l Zi . LEMMA 11. Us X is Yis Zi, where the bars denote the Zariski closures i of the respecti¨ e sets. Proof. We will prove that Us X i; the other proofs are similar. It is i sufficient to prove that X i ; U.i Given a triple Ž A, B, C . with A i-regular, the line L s Ž A, Ž1 y t . B q tA, Ž1 y t .C q tA.4 is contained in Cn3 and intersects Ui at least in the point t s 1. Since Ui is open in Cn3 and Ui l L is nonempty, Ui is dense in L. It follows that L : U.i In particular, Ž A, B, C . g U.i Remark 12. It is easy to see that Uny 1s Cn3.
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Remark 13. U1 is irreducible of dimension n2 q 2 n. This follows from the fact that X 1 is irreducible of dimension n2 q 2 n, which in turn follows from the fact that any commuting triple Ž A, B, C ., with A a 1-regular matrix, must satisfy B s f Ž A. and C s g Ž A. for uniquely determined polynomials f and g of degree at most n y 1. ŽSee w4x for instance.. We have the filtration U1: U2: ⭈⭈⭈ : Uny1s Cn3. The following is a consequence of the irreducibility of CC2 considered in the previous section. PROPOSITION 14. U2s U1, so that U2 is also irreducible of dimension n2 q 2 n. Proof. Since U1 ; U2 , it is sufficient to prove that U2; U1. By Lemma 11 above, it is sufficient to prove that X 2 : U1. Given a commuting triple Ž A, B, C ., with A a 2-regular matrix, note that there exists a point Ž D, 0. in CA2 with D a 1-regular matrix. Thus, Y1 l CA2 is nonempty Žhere, we are viewing CA2 as sitting naturally inside Cn3 .. Since CA2 is irreducible, Y1 l CA2 is all of CA2 . But Y1 l CA2; Y1s U1. Thus, Ž A, B, C . g U1. THEOREM 15. Let T be the subset of Cn3 consisting of triples Ž A, B, C . in which some two of the three matrices commute with a 2-regular matrix. For Ž A, B, C . g T, let A denote the algebra k w A, B, C x. We ha¨ e the following: Ž1.
T : U1.
Ž2. A is contained in a commutati¨ e subalgebra of MnŽ k . of dimension exactly n. Ž3.
The dimension of A is at most n.
Ž4.
The dimension of the centralizer of A is at least n.
Proof. Suppose A and B commute with a 2-regular matrix D. The line L s Ž A, B, Ž1 y t .C q tD .4 is contained in Cn3 and intersects Z2 at least in the point t s 1. Since Z2 is open in Cn3 and Z2 l L is nonempty, Z2 is dense in L. It follows that L : Z2s U2s U1, the last two equalities arising from Lemma 11 and Proposition 14. Hence Ž A, B, C . g U1. The conditions in Ž2., Ž3., and Ž4. are all polynomial conditions in the coordinates of the matrices A, B, and C. ŽThe proofs that the conditions in Ž3. and Ž4. are polynomial conditions are standard, see w8x for instance. For the condition in Ž2., see w5x.. Since these polynomial conditions hold trivially on X 1 , they hold on the closure X 1s U1. Since T ; U1, the statements immediately follow. ŽOf course, statements Ž3. and Ž4. are also direct consequences of Ž2...
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6. AN EXAMPLE We show that unlike the two-generated case Žsee w8, Theorem 1.1x., it is possible for k w A, B, C x to be of dimension n but CentŽ k w A, B, C x. to be of dimension greater than n. Let C be a nonzero nilpotent homogeneous 2-regular Žbut not 1-regular. matrix satisfying C 2 s 0. Then k w C x ( k w z xrz 2 and CentŽ C . ( Ž M2 Ž k ..w z xrz 2 . Let X 1 , X 2 , and X 3 be any three matrices in M2 Ž k . that, together with the identity matrix I2 , form a basis for M2 Ž k .. Take A s zX1 and B s zX 2 . The linear independence condition shows that k w A, B, C x is spanned by I2 , z s zI2 , zX1 , and zX 2 . On the other hand, zX 3 centralizes all four of these matrices, so dim k CentŽ k w A, B, C x. G 5. ŽIn fact, it is easy to check that dim k CentŽ k w A, B, C x. is exactly 5..
REFERENCES 1. J. Barria and P. Halmos, Vector bases for two commuting matrices, Linear and Multilinear Algebra 27 Ž1990., 147᎐157. 2. F. R. Gantmacher, ‘‘The Theory of Matrices,’’ Vol. 1, Chelsea, New York, 195. 3. M. Gerstenhaber, On dominance and varieties of commuting matrices, Ann. of Math. 73 Ž1961., 324᎐348. 4. R. Guralnick, A note on commuting pairs of matrices, Linear and Multilinear Algebra 31 Ž1992., 71᎐75. 5. R. Guralnick and B. A. Sethuraman, Commuting pairs and triples of matrices, and theorems of Motzkin᎐Taussky and Gerstenhaber, in preparation. 6. T. Laffey and S. Lazarus, Two-generated commutative matrix subalgebras, Linear Algebra Appl. 147 Ž1991., 249᎐273. 7. T. Motzkin and O. Taussky-Todd, Pairs of matrices with property L, II, Trans. Amer. Math. Soc. 80 Ž1955., 387᎐401. 8. M. Neubauer and D. Saltman, Two-generated commutative subalgebras of MnŽ F ., J. Algebra 164 Ž1994., 545᎐562. 9. A. R. Wadsworth, On commuting pairs of matrices, Linear and Multilinear Algebra 27 Ž1990., 159᎐162.