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Elementary Invariants for Centralizers of Nilpotent Matrices

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Elementary Invariants for Centralizers of Nilpotent Matrices J. Aust. Math. Soc. 86 (2009), 1–15 doi:10.1017/S1446788708000608 ELEMENTARY INVARIANTS FOR CENTRALIZERS OF NILPOTENT MATRICES JONATHAN BROWN and JONATHAN BRUNDAN ˛ (Received 7 March 2007; accepted 29 March 2007) Communicated by J. Du Abstract We construct an explicit set of algebraically independent generators for the center of the universal enveloping algebra of the centralizer of a nilpotent matrix in the general linear Lie algebra over a field of characteristic zero. In particular, this gives a new proof of the freeness of the center, a result first proved by Panyushev, Premet and Yakimova. 2000 Mathematics subject classification: primary 17B35. Keywords and phrases: nilpotent matrices, centralizers, symmetric invariants. 1. Introduction Let λ D (λ1; : : : ; λn/ be a composition of N such that either λ1 ≥ · · · ≥ λn or λ1 ≤ · · · ≤ λn. Let g be the Lie algebra glN .F/, where F is an algebraically closed field of characteristic zero. Let e 2 g be the nilpotent matrix consisting of Jordan blocks of sizes λ1; : : : ; λn in order down the diagonal, and write ge for the centralizer of e in g. g Panyushev et al. [PPY] have recently proved that S.ge/ e , the algebra of invariants for the adjoint action of ge on the symmetric algebra S.ge/, is a free polynomial algebra on N generators. Moreover, viewing S.ge/ as a graded algebra as usual so that ge is concentrated in degree one, they show that if x1;:::; xN are homogeneous generators g for S.ge/ e of degrees d1 ≤ · · · ≤ dN , then the sequence .d1;:::; dN / of invariant degrees is equal to λ1 1’s λ2 2’s λn n’s z }| { z }| { z }| { .1;:::; 1; 2;:::; 2;:::; n;:::; n/ if λ1 ≥ · · · ≥ λn, .1;:::; 1; 2;:::; 2;:::; n;:::; n/ if λ1 ≤ · · · ≤ λn. | {z } | {z } | {z } λn 1’s λn−1 2’s λ1 n’s Research supported in part by NSF grant no. DMS-0139019. c 2009 Australian Mathematical Society 1446-7887/2009 $16.00 1 Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.22, on 01 Oct 2021 at 17:03:30, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788708000608 2 J. Brown and J. Brundan [2] This is just one instance of the following conjecture formulated in this generality by Premet: for any semisimple Lie algebra g and any element e 2 g the invariant algebra g S.ge/ e is free. In [PPY] this conjecture has already been verified in many other situations besides the type A case discussed here. Returning to our special situation, let Z.ge/ denote the center of the universal enveloping algebra U.ge/. The standard filtration on U.ge/ induces a filtration on the subalgebra Z.ge/ such that the associated graded algebra gr Z.ge/ is canonically g identified with S.ge/ e ; see [D, Section 2.4.11]. We can lift the algebraically independent generators x1;:::; xN from gr Z.ge/ to Z.ge/ to deduce (without resorting to Duflo’s theorem [D, Theorem 10.4.5]) that Z.ge/ is also a free polynomial algebra. The purpose of this note is to derive an explicit formula for a set z1;:::; zN of algebraically independent generators for Z.ge/, generalizing the well-known set of generators of Z.g/ itself (the special case e D 0) that arise from the Capelli identity. We call these the elementary generators for Z.ge/. Passing back down to the associated graded algebra, one can easily obtain from them an explicit set of g elementary invariants that generate S.ge/ e . To formulate the main result precisely, we must first introduce some notation for elements of ge. Let ei; j denote the i j-matrix unit in g. Draw a diagram with rows numbered 1;:::; n from top to bottom and columns numbered 1; 2;::: from left to right, consisting of λi boxes on the ith row in columns 1; : : : ; λi , for each i D 1;:::; n. Write the numbers 1;:::; N into the boxes along rows, and use the notation row.i/ and col.i/ for the row and column number of the box containing the entry i. For instance, if λ D .4; 3; 2/ then the diagram is 1 2 3 4 5 6 7 8 9 and the nilpotent matrix e of Jordan type λ is equal to e1;2 C e2;3 C e3;4 C e5;6 C e6;7 C e8;9: For 1 ≤ i; j ≤ n and λ j − min(λi ; λ j / ≤ r < λ j , define X ei; jIr VD eh;k: (1.1) 1≤h;k≤N row.h/Di;row.k/D j col.k/−col.h/Dr The vectors fei; jIr j 1 ≤ i; j ≤ n; λ j − min(λi ; λ j / ≤ r < λ j g form a basis for ge; see [BK2, Lemma 7.3]. Write µ ⊆ λ if µ D (µ1; : : : ; µn/ is a composition with 0 ≤ µi ≤ λi for each i D 1;:::; n. Also let jµj VD µ1 C···C µn and `(µ) denote the number of nonzero parts of µ. Recall that .d1;:::; dN / are Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.22, on 01 Oct 2021 at 17:03:30, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788708000608 [3] Centralizers of nilpotent matrices 3 the invariant degrees as defined above. Given 0 6D µ ⊆ λ such that `(µ) D djµj, suppose that the nonzero parts of µ are in the entries indexed by 1 ≤ i1 < ··· < id ≤ n. Define the µth column determinant X cdet(µ) VD sgn.w/eQ I − eQ I − ::: eQ I − ; (1.2) iw1;i1 µi1 1 iw2;i2 µi2 1 iwd ;id µid 1 w2Sd where eQi; jIr VD ei; jIr − δr;0δi; j .i − 1)λi . We note by Lemma 3.8 below that all of the ei; jIr ’s appearing on the right-hand side of (1.2) necessarily satisfy the inequality λ j − min(λi ; λ j / ≤ r < λ j , so cdet(µ) is well-defined. For r D 1;:::; N, define X zr VD cdet(µ). (1.3) µ⊆λ jµjDr,`(µ)Ddr MAIN THEOREM. The elements z1;:::; zN are algebraically independent genera- tors for Z.ge/. In the situation that λ1 D···D λn, our Main Theorem was proved already by g Molev [M], following Rais and Tauvel [RT] who established the freeness of S.ge/ e in that case. Our proof for general λ follows the same strategy as Molev’s proof, but we need to replace the truncated Yangians with their shifted analogs from [BK2]. We have g also included a self-contained proof of the freeness of S.ge/ e , although as we have said this was already established in [PPY]. Our approach is similar to the argument in [RT] and different from [PPY]. One final comment. In this introduction we have formulated the Main Theorem assuming either that λ1 ≥ · · · ≥ λn or that λ1 ≤ · · · ≤ λn. Presumably most readers will prefer the former choice. However in the remainder of the article we will only actually prove the results in the latter situation, since that is the convention adopted in [BK2, BK3, BK4]. This is justified because the two formulations of the Main Theorem are simply equivalent, by an elementary argument involving twisting with an antiautomorphism of U.g/ of the form ei; j 7! −ei0; j0 C δi; j c. The remainder of the article is organized as follows. In Section 2, we derive a new ‘quantum determinant’ formula for the central elements of the shifted Yangians. In Section 3 we descend from there to the universal enveloping algebra U.ge/ to prove that the elements zr are indeed central. Then, in Section 4, we prove the freeness of g S.ge/ e by restricting to a carefully chosen slice. 2. Shifted quantum determinants The shifted Yangian Yn(σ / is defined in [BK2]. Here are some of the details. Let σ D .si; j /1≤i; j≤n be an n × n shift matrix, that is, all of its entries are nonnegative integers and si; j C s j;k D si;k whenever ji − jj C j j − kj D ji − kj. Then Yn(σ / is the Downloaded from https://www.cambridge.org/core. IP address: 170.106.33.22, on 01 Oct 2021 at 17:03:30, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788708000608 4 J. Brown and J. Brundan [4] associative algebra over F defined by generators f .r/ j ≤ ≤ g Di 1 i n; r > 0 ; f .r/ j ≤ ≤ g Ei 1 i n; r > si;iC1 ; f .r/ j ≤ ≤ g Fi 1 i n; r > siC1;i subject to certain relations. See [BK2, Section 2] for the full set. ≤ ≤ .r/ 2 For 1 i < j n and r > si; j , define elements Ei; j Yn(σ / recursively by − C .r/ VD .r/ .r/ VD T .r s j−1; j / .s j−1; j 1/U Ei;iC1 Ei ; Ei; j Ei; j−1 ; E j−1 : (2.1) ≤ ≤ .r/ 2 Similarly, for 1 i < j n and r > s j;i define elements Fi; j Yn(σ / by C − .r/ VD .r/ .r/ VD T .s j; j−1 1/ .r s j; j−1/U Fi;iC1 Fi ; Fi; j Fj−1 ; Fi; j−1 : (2.2) As in [BK3, Section 2], we introduce a new set of generators for Yn(σ /.
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