GIT-STABILITY FOR A CLASS OF LINEAR SYSTEMS
MIHAI-SORIN STUPARIU
We compute the stable and semistable points for the action of the special linear group on a certain class of linear systems with one input,nstates and no outputs.
Using coordinates, we give an explicit description of the stable and semistable locus in the casen= 2.
AMS 2000 Subject Classification: 93B27, 14L24.
Key words: stable locus, linear system, controllability.
INTRODUCTION
The aim of the present note is to study a certain class of linear systems using the tools of the Geometric Invariant Theory. Linear systems with m inputs,nstates and poutputs arise in control theory and can be described by triples of matrices
(B, A, C)∈ Mn×m(C)× Mn(C)× Mp×n(C).
One has a natural action of the group GLn(C) on the space of all linear systems, given by
g·(B, A, C) = (gB, gAg−1, Cg−1),
which corresponds to the change of states. As Byrnes and Hurt [1] have shown, controllable linear systems, that is, triples (B, A, C) such that
rk(B...AB...A2B . . ....An−1B) =n,
are precisely the stable points of this GLn(C)-action with respect to a suitable linearization. Our goal is to investigate further this kind of relationship, by computing the stable locus for the induced action of the special linear group.
In the sequel, we will restrict our attention to the case when m = 1 and p= 0. Instead of working with matrices, that is with coordinates, we fix an arbitrary n-dimensional complex vector space V. Within this framework, a linear system is given by a pair (v, ϕ) ∈ V ×End(V). We will consider a special class of such systems: the pairs (v, ϕ) such that ϕ ∈ End0(V) is a trace-free endomorphism. Our motivation is as follows: in [6] were introduced
MATH. REPORTS9(59),4 (2007), 377–384
stability concepts for oriented holomorphic pairs coupled with Higgs fields.
These are triples (E, v, ϕ) consisting of a holomorphic vector bundleE over a K¨ahler manifoldX, a holomorphic section v∈H0(E) ofE and a holomorphic section of the bundle of trace-free endomorphismsϕ ∈ H0(End0(E)). When the manifold X is only a point and the bundle reduces to a vector space, to give such a triple means, in fact, to give a pair (v, ϕ) consisting of a vector and of a trace-free endomorphism.
The aim of the first section is to prove that the stable locus for the SL(V)- action on the space V ×End0(V) coincides with the set of pairs (v, ϕ) with the property that {v, ϕ(v), . . . , ϕn−1(v)} is a basis of V, i.e., the pair (v, ϕ) is a controllable linear system. In fact, this stable locus is nothing else but the intersection of the hyperplaneV×End0(V) of the vector spaceV×End(V) and the stable locus (with respect to a suitable linearisation) of the GL(V)-action on the whole space of linear systems.
In the second part of the paper, we will restrict our attention to the case when the dimension of the complex vector space V is two. We study first, using coordinates, the structure of the (semi)stable locus in this particular case and the intersection of these loci with the set of pairs (v, ϕ) such that ϕ(v) = 0. We obtain subsets ofC5 related to the vanishing loci of some homo- geneous polynomials and describe their images under the natural projection map fromC5 to P4.
1. ANALYSIS OF THE STABILITY
LetGbe a connected reductive complex Lie group and let furtherρ:G→ GL(W) be a representation ofGon a finite dimensional complex vector space W. As noticed in [3], any character χ : G → C∗ of G yields a linearization of the trivial complex line bundle L over W. On the total space of L−1 = W ×C, the group G acts by g·(w, λ) := (ρ(g)w, χ−1(g)λ). In this context, Mumford’s definitions for (semi)stability with respect to this linearization, given in [4], can be phrased as follows [3]. A point w ∈ W is called χ- semistableif there existn≥1 and aχn-equivariant polynomialf ∈C[W]G,χn such thatf(w)= 0. If, moreover, theG-operation on{w|f(w)= 0}is closed and the dimension of the stabilizer Gw of w is equal to the dimension of the kernel of the representation ρ, then w will be called χ-stable. A very useful tool for the computation of the (semi)stable points is the Hilbert criterion [4], [3], which asserts that a pointw∈W is:
(i) not stable with respect to the given action if and only if there exists a one-parameter subgroupλ:C∗→Gsuch that the limit
(1.1) lim
z→∞λ(z)·w exists,
(ii) not semistable with respect to the given action if and only if there exists a one-parameter subgroupλ:C∗ → G such that the limit (1.1) exists and is equal to 0.
As we noticed, our goal is to describe the sets of stable and semistable points for a particular representation. More precisely, letV be ann-dimensional complex vector space. We consider the representation ρ of the group SL(V) on the spaceV ×End0V given by
(1.2) g·(v, ϕ) := (gv, g◦ϕ◦g−1).
In this case, since one has to consider only the trivial character χ0, we will simply say (semi)stable instead of χ0-(semi)stable. The aim of the present section is to prove the following result.
Proposition1.1. LetV be ann-dimensional complex vector space. Take the action of SL(V) on the space V ×End0V given by (1.2). An element (v, ϕ)∈V ×End0V is:
i) stableif and only if v∧ϕ(v)∧ · · · ∧ϕn−1(v)= 0,
ii) semistable if and only if v∧ϕ(v)∧ · · · ∧ϕn−1(v)= 0 or ϕn= 0.
Throughout this paper the following notation is used: for vectorsv1, . . . , vr ∈V, we write
v1∧ · · · ∧vr = 0 if the vectorsv1, . . . , vr are linearly dependent;
v1∧ · · · ∧vr = 0 if the vectorsv1, . . . , vr are linearly independent.
Let now λ:C∗ → SL(V) be a non-trivial one-parameter subgroup. As stated, for instance, in [2, 2.1], there exists an ordered basis B= (b1, . . . , bn) ofV and weights γ1, γ2, . . . , γn−1, γn ∈Zwith
γ1 ≥ · · · ≥γn−1 ≥γn and such that for anyz∈Cwe have
λ(z)·bi =zγibi, i= 1, . . . , n.
Moreover, sinceλ(z)∈SL(V), we deduce thatγn=−γ1−· · ·−γn−1and, since λis not trivial, the inequalities γ1 >0> γn hold. The action of SL(V) on V induces a canonical action of SL(V) on V∨. For the one-parameter subgroup λone has
λ(z)·b∨i =z−γib∨i, i= 1, . . . , n.
In this way, one gets an action ofλon the basis
(1.3) B= (b1⊗b∨1 −bn⊗b∨n, . . . , bn−1⊗b∨n−1−bn⊗b∨n, . . . ,(bj⊗b∨i)i=j, . . .) of End0V. The next results follow at once.
Lemma 1.2. Let V be an n-dimensional complex vector space and let λ:C∗→SL(V) be a non-trivial one-parameter subgroup. There exists a basis B= (b1, . . . , bn) of V and integers γ1, . . . , γn−1, not all of them zero, with
γ1≥γ2 ≥ · · · ≥γn−1 ≥ −γ1− · · · −γn−1 such that the action ofλ(z) is given by
(1.4)
b1 →zγ1b1, . . . , bn−1→zγn−1bn−1, bn→z−γ1−···−γn−1bn; bi⊗b∨i −bn⊗b∨n →bi⊗b∨i −bn⊗b∨n, i= 1, . . . , n−1;
bi⊗b∨j →z−γj+γi(bi⊗b∨j), i, j = 1, . . . , n−1, i=j;
bi⊗b∨n →zγ1+···+2γi+···+γn−1(bi⊗b∨n), i= 1, . . . , n−1;
bn⊗b∨i →z−γ1−···−2γi−···−γn−1(bn⊗b∨i), i= 1, . . . , n−1.
Lemma 1.3. Let γ1, . . . , γn−1 be as in Lemma 1.2. There exist integers r≥s≥1, such that
γr≥0, γr+1 <0,
γ1 =γ2 =· · ·=γs> γs+1≥ · · · ≥γn−1 ≥ −γ1− · · · −γn−1. Moreover,
i) γ1+· · ·+ 2γi+· · ·+γn−1 >0 for i= 1, . . . , s,
ii) γi−γj >0 for i∈ {1, . . . , s}, j∈ {s+ 1, . . . , n−1}.
Take now a point (v, ϕ) ∈V ×End0V. Suppose that an ordered basis B = (b1, . . . , bn) of V is fixed. By (1.3) we obtain an ordered basis B of End0V. Using the frame (B,B), we introduce coordinates inV ×End0V and regard the pair (v, ϕ) as an element
(v1, . . . , vn, ϕ11, . . . , ϕn−1n−1, . . . ,(ϕij)i=j, . . .)∈Cn2+n−1. Note that forϕas above the associated matrix with respect to Bis
ϕB =
ϕ11 · · · ϕ1n
· · · · · · ·
ϕn1 · · · −ϕ11− · · · −ϕn−1n−1
.
Lemma 1.4. Let V be an n-dimensional complex vector space. Take the action(1.2) of SL(V) on V ×End0V and fix(v, ϕ)∈V ×End0V.
i) There exists a one-parameter subgroup λ : C∗ → SL(V) such that the limit
(1.5) lim
z→∞λ(z)·(v, ϕ) exists if and only ifv∧ϕ(v)∧ · · · ∧ϕn−1(v) = 0.
ii) There exists a one-parameter subgroup λ:C∗ →SL(V) such that the limit(1.5) exists and is equal to (0,0) ∈V ×End0V if and only if v∧ϕ(v)∧
· · · ∧ϕn−1(v) = 0 and ϕn= 0.
Proof. i) “⇒”: Take an ordered basis B of V associated with λ as in Lemma 1.2 and integers r and s as in Lemma 1.3. Using the frame B, introduce coordinates inV ×End0V. By the assumption that the limit (1.5) exists, we have
v1=· · ·=vs = 0, ϕij = 0 ∀i= 1, . . . , s; ∀j=s+ 1, . . . , n.
For anyk= 1, . . . , n−1 we decompose the vectorϕk(v) with respect toB as ϕk(v) = n
j=1
(ϕk(v))jbj.
We deduce that its firstscomponents are equal to zero, i.e., (ϕk(v))1 = 0, . . . , (ϕk(v))s= 0, hencev∧ϕ(v)∧ · · · ∧ϕn−1(v) = 0.
“⇐”: In the casev= 0, take an ordered basisB= (b1, . . . , bn) of V such that the associated matrixϕB is lower triangular. Putλ:C∗ →SL(V),
λ(z)·b1 :=zn−1b1, λ(z)·b2:=z−1b2, . . . , λ(z)·bn:=z−1bn. By the assumption thatϕB is lower triangular,ϕhas the form
ϕ=
n−1 i=1
ϕii(bi⊗b∨i −bn⊗b∨n) +
i>j
ϕijbi⊗b∨j.
Hence
λ(z)ϕ=
n−1 i=1
ϕii(bi⊗b∨i −bn⊗b∨n) +
i>j
ϕijzγi−γj(bi⊗b∨j),
and the limit
z→∞lim λ(z)ϕ exists. Therefore, the limit (1.5) exists.
In the casev= 0, one can findq ∈ {1, . . . , n−1} such that v∧ϕ(v)∧ · · · ∧ϕq−1(v)= 0 and v∧ϕ(v)∧ · · · ∧ϕq(v) = 0.
Sets:=n−q and by our assumption we haves≥1. Putting bs+1:=v, bs+2:=ϕ(v), . . . , bn:=ϕq−1(v)
one can find vectors b1, . . . , bs ∈ V such that B = (b1, . . . , bn) is a frame of V with the property that the associated matrix ϕB ofϕ is a lower triangular
matrix. Define now the one-parameter subgroup λ :C∗ → SL(V) such that its action onB is given by
λ(z)·b1 :=zn−sb1, . . . , λ(z)·bs:=zn−sbs, λ(z)·bs+1 :=z−sbs+1, . . . , λ(z)·bn:=z−sbn.
Using the fact that the firstscomponents ofv(with respect toB), on whichγ acts by positive weights, are equal to zero, we deduce that limzλ(z)v= 0. On the other hand, sinceϕB is lower triangular, the limit limzλ(z)ϕ also exists.
Moreover, denoting byψ this limit, the entries of its associated matrix with respect toB are (ψB)ii= (ϕB)iifor anyi= 1, . . . , sand (ψB)jk= 0 otherwise.
ii) “⇒”: If λ:C∗ → SL(V) is a one-parameter subgroup such that the limit (1.5) is (0,0), using again Lemmas 1.2 and 1.3 we can find an ordered basis B of V such that with respect to this frame, ϕB is a lower triangular matrix. Moreover, since the vectors bi⊗b∨i −bn⊗b∨n are invariant under the action ofλ(z), the entries of the main diagonal of ϕmust be 0, hence we have ϕn= 0. The other condition was already obtained in i).
“⇐”: The construction is analogous to i). Additionally, the fact that ϕn = 0 implies that we have 0 on the main diagonal of the matrixϕB, hence ψB = 0. Consequently, limzλ(z)·(v, ϕ) = 0.
The statements of Proposition 1.1 follow now directly from Lemma 1.4 and from the Hilbert Criterion.
Remark. As we noticed, the case of the natural GL(V)-action on the spaceV×End(V) was studied in [1], where the authors proved that the set of pairs (v, ϕ) with the propertyv∧ϕ(v)∧ · · · ∧ϕn−1(v)= 0 coincides with both sets ofχ1-stable and χ1-semistable points, whereχ1 = det. Intersecting this set with the hyperplaneV ×End0(V) one obtains the set of χ0-stable points of the SL(V)-action on V ×End0(V).
2. THE CASE OF DIMENSION TWO
In this section we restrict our attention to the case where dimCV = 2.
We start by fixing an ordered basisB= (b1, b2) of V and consider the induced one ofV ×End0V, namely,
(b1, b2; b1⊗b∨1 −b2⊗b∨2, b2⊗b∨1, b1⊗b∨2).
Introducing coordinates with respect to this frame, a pair (v, ϕ) can be re- garded as a point (x, y;a, b, c) ∈C5, where
ϕB=
a b c −a
is the matrix associated withϕwith respect to the frame B. We haveϕ2 = 0 if and only if detϕ = 0, that is, a2+bc = 0. On the other hand, one has ϕ(v) = (ax+by, cx−ay) and the condition v∧ϕ(v) = 0 is equivalent to
det
x y ax+by cx−ay
= 0 ⇐⇒∆ = 0.
Since both polynomials a2+bcand cx2−2axy−by2 are homogeneous, it is natural to consider the canonical projection π : C5\ {0} →P4 and the varieties Z(δ)⊂P4 and Z(∆)⊂P4, where
δ= det
a b c −a
, ∆ = det
a b x c −a y
−y x 0
.
Hence, for (v, ϕ)∈C5\ {0}we havev∧ϕ(v) = 0 if and only ifπ(v, ϕ)∈Z(∆) andϕ2 = 0 if and only ifπ(v, ϕ)∈Z(δ). In this way, we obtain a stratification ofP4:
P4 = (P4\Z(∆))(Z(∆)\Z(δ))(Z(∆)∩Z(δ)).
The strata correspond, via π, to the sets of stable, properly semistable, re- spectively non semistable points.
Our next goal is to study the intersection of these sets with the subvariety Nˆ :={(v, ϕ)∈V ×End0V |ϕ(v) = 0}.
We first notice that there are no stable points in ˆN. In fact, ˆN decomposes as (2.1) Nˆ = ˆNssNˆnon−ss,
where
Nˆss={(v, ϕ) ∈Nˆ |detϕ= 0}={(0, ϕ)|detϕ= 0}, Nˆnon−ss={(v, ϕ)∈Nˆ |detϕ= 0}.
Introduce the set
N :={[x:y:a:b:c]∈P4|ax+by= 0, cx−ay= 0}.
Clearly, ˆN ⊂ C5 is the cone over N. Our goal is to describe explicitely the intersection of N with the above strata of P4. The first remark we make is thatN ⊂Z(∆), i.e., there are no stable points in N. Put
Nss:=π( ˆNss), Nnon−ss:=π( ˆNnon−ss\ {0}).
These sets are the intersection ofN with the strataZ(∆)\Z(δ), respectively Z(∆)∩Z(δ). Consider now the projective plane
P :={[0 : 0 :a:b:c]∈P4|[a:b:c]∈P2} P2,
and the family Σ of surfaces defined as
Σ ={[x:y:a:b:c]∈P4 |a2+bc= 0, ax+by= 0, cx−ay= 0}=
=
[x:y:a:b:c]|rk
a b x c −a y
≤1 ,
which is the blow up ofP2 at a point. These surfaces intersect along the conic C:={[0 : 0 :a:b:c]|[a:b:c]∈P2, a2+bc= 0}.
We have
Nss=P \ C, Nnon−ss= Σ,
that isNis the union of the projective planePand the blow up ofP2at a point.
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[6] M.S. Stupariu,The Kobayashi-Hitchin correspondence for vortex-type equations coupled with Higgs fields. I. Math. Reports3(53)(2001),4, 425–444.
Received 21 May 2007 University of Bucharest
Faculty of Mathematics and Computer Science Str. Academiei 14
010014 Bucharest, Romania
