REAL VARIETIES
TOBIAS FRIEDL, CORDIAN RIENER, AND RAMAN SANYAL
Abstract. LetXbe a nonempty real variety that is invariant under the action of a reflection groupG. We conjecture that ifXis defined in terms of the firstkbasic invariants ofG(ordered by degree), thenX meets a k-dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most3, and F4 and we give computational evidence forH4. This is a generalization of Timofte’s degree principle to reflection groups. For general reflection groups, we compute nontrivial upper bounds on the minimal dimension of flats of the reflection arrangement meetingX from the combinatorics of parabolic subgroups. We also give generalizations to real varieties invariant under Lie groups.
1. Introduction
A real variety X ✓Rn is the set of real points simultaneously satisfying a system of polynomial equations with real coefficients, that is,
X = VR(f1, . . . , fm) := {p2Rn:f1(p) =f2(p) =· · ·=fm(p) = 0},
for somef1, . . . , fm2R[x] :=R[x1, . . . , xn]. In contrast to working over an algebraically closed field, the question ifX 6=?is considerably more difficult to answer, both theoretically and in practice;
see [3]. Timofte [18] studied real varieties invariant under the action of the symmetric groupSn, and proved an interesting structural result. A Sn-invariant variety can be defined in terms of symmetricpolynomials, that is, polynomialsf2R[x]such thatf(x⌧(1), . . . , x⌧(n)) =f(x1, . . . , xn) for all permutations⌧ 2Sn. Recall that the fundamental theorem of symmetric polynomials states that a polynomial f is symmetric if and only if f is a polynomial in the elementary symmetric polynomials e1, . . . , en. Let us call aSn-invariant varietyX k-sparseif X = VR(f1, . . . , fm) for some symmetric polynomialsf1, . . . , fm2R[e1, . . . , ek].
Theorem 1 ([18]). Let X ✓Rn be a nonemptySn-invariant real variety. If X isk-sparse, then there is a point p2X with at mostk distinct coordinates.
Viewing the symmetric groupSnas a reflection group inRnyields a sound geometric perspective on this result: As a group of linear transformations,Snis generated by reflections in the hyperplanes Hij ={p:pi =pj}for1i < j n. The ambient spaceRn is stratified by the arrangement of reflection hyperplanesH={Hij:i < j}. The closed strataHk✓Rnare the intersections ofn k linearly independent reflection hyperplanes. Timofte’s result then states that a k-sparse varietyX is nonempty if and only ifX\Hk6=?. Such a point of view can be taken for general real reflection groups and the aim of this paper is a generalization of Theorem1.
Date: January 19, 2018.
2010Mathematics Subject Classification. 14P05, 14P10, 20F55.
Key words and phrases.reflection groups, reflection arrangements, invariant real varieties, real orbit spaces.
T. Friedl and R. Sanyal were supported by the DFG-Collaborative Research Center, TRR 109 “Discretization in Geometry and Dynamics”. T. Friedl received additional funding from a scholarship of the Dahlem Research School at Freie Universität Berlin.
81
A (real) reflection group G acting on V ⇠= Rn is a finite group of orthogonal transformations generated by reflections. The reflection group G is irreducibleif G is not the product of two nontrivial reflection groups. Associated toGis itsreflection arrangement
H = H(G) := {H= kerg:g2Greflection}.
TheflatsofHare the linear subspaces arising from intersections of hyperplanes inH. The arrange- ment of linear hyperplanes stratifiesV with strata given by
Hi = Hi(G) := {p2V :pis contained in a flat of dimensioni}.
In particular,Hn=V. We callGessentialifGdoes not fix a nontrivial linear subspace or, equiv- alently, ifH0={0}. IfGis essential, then therankofGisrank(G) := dimV. Reflection groups naturally occur in connection with Lie groups/algebras and are well-studied from the perspective of geometry, algebra, and combinatorics [11,5,4]. A complete classification of reflection groups can be given in terms of Dynkin diagrams (see [11]). There are four infinite families of irreducible reflection groupsSn⇠=An 1, Bn, Dn, I2(m)and six exceptional reflection groupsH3, H4, F4, E6, E7,andE8. The linear action of G on V induces an action on the symmetric algebra R[V] ⇠= R[x1, . . . , xn] by g·f(x) :=f(g 1·x). Chevalley’s Theorem [11, Ch. 3.5] states that the ring R[V]G of poly- nomials invariant under G is generated by algebraically independent homogeneous polynomials
⇡1,⇡2, . . . ,⇡n 2 R[V]. The collection ⇡1, . . . ,⇡n is called a set of basic invariantsfor G. The basic invariants are not unique, but their degreesdi(G) := deg⇡iare. Throughout, we will assume that the basic invariants are labelled such that d1d2· · ·dn. In accord withSn-invariant varieties, we call aG-invariant varietyX= VR(f1, . . . , fm)k-sparseiff1, . . . , fmcan be chosen in R[⇡1, . . . ,⇡k]for some choice of basic invariants⇡1, . . . ,⇡nordered by nondecreasing degrees. The following is the main result of the paper.
Theorem 2. LetGbe a reflection group of typeI2(m), An 1, Bn, Dn, H3,orF4andX a nonempty G-invariant real variety. IfX isk-sparse, thenX\Hk(G)6=?.
Since the first basic invariant of an essential reflection group is a scalar multiple ofp2(x) =kxk2, Theorem2is trivially true for reflection groups of rank2. The infinite familiesAn 1, Bn, and Dn are treated in Section 2. Timofte’s original proof and its simplification given by the second author in [15] use properties of the symmetric group that are not shared by all reflection groups (such asDn) and we highlight this difference in Example1and Remark1. In Section3, we prove the following general result that implies Theorem2in the casek=n 1.
Theorem 3. LetGbe an essential reflection group of ranknand X= VR(f1, . . . , fm)nonempty.
If there isj2{1, . . . , n}suchf1, . . . , fm2R[⇡i:i6=j], thenX\Hn 1(G)6=?.
In particular, this result yields Theorem2 for all reflection groups of rank 3. The group F4 is treated in Section3and we provide computational evidence that Theorem2also holds forH4. That supports the following conjecture.
Conjecture 1. Let G be an irreducible and essential reflection group. Then any nonempty and k-sparseG-invariant real varietyX intersectsHk(G).
Proposition8provides a different geometric perspective on Conjecture1in terms of real orbit spaces and implies (Proposition9) thatX will in general not meetHl(G)forl < k. In Section4, we prove a weaker form of Conjecture 1under an extra assumption on the defining polynomials of X. In Section5, we obtain upper bounds on the dimension of the stratum that meetsX in terms of the combinatorics of parabolic subgroups ofG. Our results generalize to varieties invariant under the adjoint action of Lie groups and we explore this connection in Section6.
Acevedo and Velasco [1] independently considered the related problem of certifying nonnegativ- ity of G-invariant homogeneous polynomials. They show that low-degree forms (where the exact
degree depends on the group) are nonnegative if and only if they are nonnegative on Hn 1(G).
Questions of nonnegativity of polynomials f 2 R[V]G are subsumed by our results. Let us call a G-invariant semialgebraic setS✓V k-sparse ifS is defined in terms of equations and inequalities with polynomials inR[⇡1, . . . ,⇡k].
Proposition 4. LetGbe a reflection group for which Conjecture1holds. LetS✓V be ak-sparse semialgebraic set and letf2R[⇡1, . . . ,⇡k]. Thenf is nonnegative/positive onS if and only iff is nonnegative/positive onHk(G)\S.
Proof. IfS isk-sparse, then theG-invariant variety
(1) Xk(q) := {p2V :⇡i(p) =⇡i(q)fori= 1, . . . , k}
is contained inS for anyq2S. Assume that there is a pointq2S withf(q)<0. By assumption f =F(⇡1, . . . ,⇡k)for someF 2R[y1, . . . , yk]. Hencef is negative (and constant) onXk(q) ✓S.
By constructionXk(q)isk-sparse and, sinceGsatisfies Conjecture1,Xk(q)\Hk(G)6=?. ⇤ The proof of Proposition4makes use of a key observation: It suffices to consider invariant varieties of the form (1) as anyk-sparse varietyX containsXk(q)for allq2X. We callXk(q)aprincipal k-sparse variety. Lastly, let us emphasize again that we will work with real varieties exclusively.
In particular, set-theoretically, every real varietyX = VR(f1, . . . , fm)is the set of solutions to the equationf(x) = 0forf=f12+f22+· · ·+fm2.
Acknowledgements. We are much indebted to Christian Stump for the many helpful discussions regarding the combinatorics of reflection groups and their invariants. We also thank Florian Frick and Christian Haase for an interesting but fruitless afternoon of orbit spaces. We also thank Mareike Dressler for help with GloptiPoly and Vic Reiner for suggesting Chevalley’s Restriction Theorem.
2. The infinite familiesAn 1, Bn,andDn
In this section, we prove Theorem 2 for the reflection groups of type An 1, Bn, and Dn. The symmetric group Sn acts onRn but is not essential as it fixes R1. The restriction to {x2 Rn: x1+· · ·+xn= 0}is the essential reflection group of typeAn 1. The reflection arrangementH(Sn) was described in the introduction. Thek-stratumHk(Sn)is given by the pointsp2Rnthat have at mostk distinct coordinates. A set of basic invariants is given by theelementary symmetric polynomials
ek(x) := X
1i1<···<ikn
xi1· · ·xik
or, alternatively, by thepower sums
sk(x) := xk1+xk2+· · ·+xkn,
fork= 1, . . . , n. The groupBn=SnoZn2 acts onV =Rnbysigned permutations with reflection hyperplanes {xi =±xj} and {xi = 0}for1 i < j n. A point plies in Hi(Bn) if and only if (|p1|, . . . ,|pn|) has at mostidistinct nonzero coordinates. A set of basic invariants is given by
⇡i(x) =s2i(x) =si(x21, . . . , x2n). The index-2subgroup Dn ofBn given by the semidirect product ofSn with ‘even sign changes’ yields a reflection group with reflection hyperplanes{xi=±xj}for 1i < jn. Thek-stratum ofDnis a bit more involved to describe: denote byM the set of all p2Rnwith exactly one zero coordinate. Then
(2) Hk(Dn) = (Hk(Bn)\M)[(Hk 1(Bn)\M).
The invariant that distinguishesDnfromBnis given byen(x) =x1x2· · ·xn. A set of basic invariants forDnare⇡1(x), . . . ,⇡n(x)with
(3) ⇡k(x) :=
8>
<
>:
s2k(x) for1k bn2c, en(x) fork=bn2c+ 1, and s2k 2(x) forbn2c+ 1< kn.
We start with the verification of Theorem2forSnwhich is exactly Theorem1. The proofs forBn
andDnwill rely on the arguments forAn 1.
Proof of Theorem2forAn 1⇠=Sn. Let us first assume that⇡i(x) =si(x) =xi1+· · ·+xinare the power sums for i= 1, . . . , n. It suffices to show the claim for a principalk-sparse varietyXk(p0) as defined in (1), i.e. thatXk(p0)\Hk(Sn)6=? forp02 X. Since ⇡2(p) = kpk2, we conclude thatXk(p0)is compact and⇡k+1attains its maximum overXk(p0)in a pointq. At this point, the JacobianJacq(⇡1, . . . ,⇡k+1) = rs1(q), . . . ,rsk+1(q) has rank< k+ 1. We claim thatq2Hkif and only if the JacobianJ= Jacq(⇡1, . . . ,⇡k+1)has rank< k+ 1. Indeed, up to scaling columns, J is given by
0 BB B@
1 1 · · · 1 q1 q2 · · · qn
... ...
q1k q2k · · · qkn 1 CC CA.
The k+ 1minors are thus Vandermonde determinants all of which vanish if and only if q2 Hk, by the description ofk-strata forSn. For an arbitrary choice of basic invariants, the result follows
from Lemma5below. ⇤
Lemma 5. Fix a reflection groupGacting onV. Let⇡1, . . . ,⇡nand⇡01, . . . ,⇡0nbe two sets of basic invariants and let1knsuch thatdk+1> dk. ThenR[⇡1, . . . ,⇡k] = R[⇡01, . . . ,⇡0k].Moreover, rank Jacp(⇡1, . . . ,⇡k) = rank Jacp(⇡10, . . . ,⇡k0)for allp2V.
Proof. For every1ik,⇡i=Fi(⇡01, . . . ,⇡0n) for some polynomialFi(y1, . . . , yn). Homogeneity and algebraic independence imply thatFi2R[y1, . . . , yk]. This shows the inclusionR[⇡1, . . . ,⇡k]✓ R[⇡01, . . . ,⇡0k]. Note that
Jacp(⇡1, . . . ,⇡k) = Jac⇡0(p)(F1, . . . , Fk)·Jacp(⇡01, . . . ,⇡0k)
for every p 2 V. The same argument applied to ⇡0i now proves the first claim and shows that Jac⇡0(p)(F1, . . . , Fk)has full rank and this proves the second claim. ⇤ We proceed to the reflection groups of typeBn.
Proof of Theorem2forBn. By Lemma5and the fact that the degreesdi(Bn)are all distinct, we may assume that⇡i(x) =s2i(x)fori= 1, . . . , n. Moreover, we can assume that X is a principal k-sparse variety, that is,
X = Xk(p) = {x2Rn:s2i(x) =s2i(p)fori= 1, . . . , k}.
SinceXk(p) =Xk(q)for allq2Xk(p), we can assume thatp= (p1, . . . , pr,0, . . . ,0)2X with the property thatp1· · ·pr6= 0andris minimal.
If r =n, then X does not meet any of the coordinate hyperplanes {xi = 0}. Let q 2X be an extreme point of ⇡k+1 overX. At this point, the JacobianJ = Jacq(⇡1, . . . ,⇡k+1) does not have
full rank and hence every maximal minor of
J =
0 B@
q1 q2 · · · qn
... ...
q12k 1 q22k 1 · · · qn2k 1 1 CA
vanishes. Since qi 6= 0for alli= 1, . . . , n, the Vandermonde formula implies that(q21, q22, . . . , qn2) has at mostkdistinct coordinates, which yields the claim.
Ifr < n, we can restrictX to the linear subspaceU ={x2Rn:xr+1=· · ·=xn= 0}⇠=Rr. The set X0 :=X\U ✓Rr is nonempty and, in particular, a k-sparseBr-invariant variety that stays away from the coordinate hyperplanes in Rr. By the previous case, there is a pointq0 2X0 such that(|q01|, . . . ,|qr0|)has at mostkdistinct coordinates. By construction,q= (q0,0)2X\Hk(Bn),
which proves the claim. ⇤
The key to the proof of Theorem 2forAn 1and Bn is the strong connection between the strata Hkand the ranks of the JacobiansJac(⇡1, . . . ,⇡k+1).
Corollary 6. LetG2{Sn, Bn}and⇡1, . . . ,⇡na set of basic invariants forG. Then a pointq2V lies inHk(G)for0kn 1if and only ifJacp(⇡1, . . . ,⇡k+1) has rank at mostk.
It is tempting to believe that such a statement holds true for all reflection groups and, indeed, necessity follows from a well-known result of Steinberg [17]. However, the following example shows that Corollary6does not hold in general.
Example 1. Consider the groupG=D5acting onR5and the pointp= (1,1,1,1,0). The point lies inH2(D5)\ H1(D5), that is,plies on exactly3linearly independent reflection hyperplanes. On the other hand, for any choice of basic invariants⇡1, . . . ,⇡5the gradientsrp⇡1,rp⇡2are linearly dependent. Indeed, for⇡1=kxk2=x21+· · ·+x25and⇡2=x41+· · ·+x45, this is easy to check and this extends to all choices of basic invariants using Lemma5.
Remark 1. Example1also serves as a counterexample to generalizations of Corollary6to all finite reflection groups considered in [6, Lemma 1’] (without a proof) and [2, Statement 3.3]. Moreover, in the language of Acevedo and Velasco [1, Definition 7], it is the first example of a reflection group not satisfying theminor factorization condition.
The following proof of Theorem2for typeDndoes not rely on an extension of Corollary6.
Proof of Theorem2forDn. Let ⇡1, . . . ,⇡n be a choice of basic invariants for Dn and let X = Xk(q) ✓Rn for someq 2Rn and 1k < n. Ifnis odd or if k6=bn2c, then, by Lemma5, we can assume that the basic invariants are given by (3). Ifnis even andk=bn2c, then⇡bn2c(x) =
↵sn(x)+ en(x),for some 6= 0. We can also assume thatq= (q1, . . . , ql,0, . . . ,0)withq1· · ·ql6= 0 andlmaximal among all points inXk(q). We distinguish two cases.
Case l < n: In this case, en(x)is identically zero on Xk(q) and X0 :=Xk(q)\{x :xn = 0}is nonempty. Ifk bn2c+ 1, then we can identify
X0 = {x02Rn 1:s2i(x0,0) =s2i(q)fori= 1, . . . , k 1}.
HenceX0 is a real variety in Rn 1 invariant under the action of Bn 1 and X0 is (k 1)-sparse.
By Theorem2forBn 1,X0\Hk 1(Bn 1)6=?. The claim now follows the description ofHk(Dn) given in (2).
If k < n2, consider the Jacobian of⇡1=s2, . . . ,⇡k=s2kand the(l+ 1)-th elementary symmetric polynomialel+1(x)atq
(4) J = Jacq(s2, . . . , s2k, el+1) = 0 BB BB B@
q1 q2 · · · ql 0 0 · · · 0 q13 q23 · · · ql3 0 0 · · · 0
... ... ... ... ... ... ...
q2k1 1 q2k2 1 · · · ql2k 1 0 0 · · · 0 0 0 0 0 q1· · ·ql 0 · · · 0 1 CC CC CA .
We observe that the(l+ 1)-th elementary symmetric functionel+1(x)is identically zero onXk(q) and hence the gradients of⇡1, . . . ,⇡k andel+1are linearly dependent onXk(q). In particular, the JacobianJ has rankk. Sinceq1· · ·ql6= 0, the Vandermonde minors imply
Y
i,j2I,i<j
qi2 q2j = 0
for anyI✓{1, . . . , l}with|I|=k. This shows thatq2Hk 1(Bn)✓Hk(Dn).
If k = n2, then n is even and Xk(q) is cut out by s2, . . . , sn 2 and possibly sn, since en(x) is identically zero onXk(q). Thus the argument above remains valid.
Casel=n: Ifk < n2, setf :=en. Ifk bn2c+ 1, setf:=s2k. For the special case thatneven and k= n2, we setf =enif↵6= 0andf =snotherwise. Letr2Xk(q)be a maximizer of|f(x)|. In particular,r1· · ·rn6= 0. Up to row and column operations, the JacobianJ= Jacq(s2, s4, . . . , s2k, f) is of the form
(5) J =
0 BB BB B@
r1 r2 · · · rn
r31 r23 · · · rn3
... ... ...
r2k1 1 r2k2 1 · · · r2kn 1 b
r1r2· · ·rn r1br2· · ·rn · · · r1r2· · ·brn 1 CC CC CA ,
wherebriis to be omitted from the product. Multiplying thei-th column byriand dividing the last row by r1· · ·rn, we get a Vandermonde matrix of rankk. Hence (|r1|, . . . ,|rk|)has at most k distinct entries. Since all entries are nonzero, it follows thatr2Hk(Dn). ⇤ The proof actually gives stronger implications for theBn-case.
Corollary 7. Let1k n 2. Then every nonempty Bn-invariant, k-sparse varietyX meets Hk(Dn).
Proof. ForX =Xk(q), we can assume thatq= (q1, . . . , ql,0, . . . ,0)withqi6= 0for1iland lmaximal. Iflk, thenq2Hk(Dn)and we are done. So assumek < ln. We distinguish two cases: If ln 1, let f =el+1 and r2 Xk(q) arbitrary. Ifl=n, letf =en andr2Xk(q)a maximizer of|f|. The corresponding Jacobians (4) and (5) fors2, . . . , s2k, fatryield the claim. ⇤
3. Real orbit spaces and reflection arrangements
The reflection arrangementHdecomposesV into relatively open polyhedral cones. The closure of a full-dimensional cone in this decomposition serves as afundamental domain: For everyp2V the orbitGpmeets in a unique point; see [11, Thm. 1.12]. On the other hand, the basic invariants define anorbit map⇡:V !Rngiven by⇡(x) = (⇡1(x), . . . ,⇡n(x)). The basic invariants separate orbits, that is, ⇡(p) = ⇡(q) if and only if q 2 Gp for all p,q 2 V. The image S := ⇡(V) is homeomorphic toV /Gand, by abuse of terminology, we callSthereal orbit space. Since⇡is an
algebraic map, S is semialgebraic (with an explicit description given in [14]). Restricted to the map⇡| : !S is a homeomorphism, by [14, Prop. 0.4]. Moreover,
(6) ⇡ 1(@S) = [
p2@
Gp = Hn 1(G),
where @S denotes the boundary ofS and where the second equality follows from [11, Thm. 1.12].
Observe that neither the orbit spaceS nor the fundamental domain are uniquely determined by G.
In terms of the orbit map, Conjecture 1 can be put in a more general context. For J ✓ [n] :=
{1, . . . , n}, let us write⇡J(x) = (⇡i(x) :i2J). For givenJ, we can ask for the smallest0tn such that⇡J(V) = ⇡J(Ht).
Proposition 8. LetG be an irreducible and essential reflection group. Then Conjecture1is true forGif and only if forJ={1, . . . , k}
⇡J(V) = ⇡J(Hk).
Proof. Forq2V, we haveXk(q) =⇡J1(⇡J(q)). Hence,Xk(q)\Hk6=?forq2V if and only if
there is somep2Hksuch that⇡J(q) =⇡J(p). ⇤
A generalization of Theorem1toJ-sparsesymmetric polynomialsf 2R[⇡i:i2J]was considered in [16]. The correspondence given in Proposition 8 also shows that the dimensions of strata in Conjecture1are best possible.
Proposition 9. LetJ✓[n]and0tn such that⇡J(V) = ⇡J(Ht). Thent |J|.
Proof. The set⇡J(V)is the projection of the real orbit spaceS onto the coordinates indexed byJ and hence is of full dimension|J|. By invariance of dimension, this implies thatt= dimHt |J|. ⇤ For the next result recall that, by definition,G⇢O(V)and hencekxk2=hx,xiis an invariant of G.
Lemma 10. LetGbe a finite reflection group and⇡1, . . . ,⇡n a choice of basic invariants such that
⇡i(x) =kxk2 for somei. Then the orbit spaceS =⇡(V)is line-free, that is, ifL✓V is an affine subspace such that L✓S, thenLis a point.
Proof. Since⇡i(x) =kxk2 0for allx2V, the linear function`(y) =yiis nonnegative onS⇢Rn. Hence, ifL✓S is an affine subspace, then`is constant onL. Let ✓V be a fundamental domain forG. ThenL=S\L is homeomorphic toLˆ:={p2 :kpk2=c}for somec 0. This implies
thatLis compact which proves the claim. ⇤
The following result is a slightly stronger but more technical extension of Theorem3.
Theorem 11. Let Gbe an essential reflection group with a choice of basic invariants ⇡1, . . . ,⇡n. Let f 2R[V]G be an invariant polynomial such that f is at most linear in ⇡k for some k. Then VR(f)6=?if and only ifVR(f)\Hn 16=?.
Iff1, . . . , fmare invariant polynomials that do not depend on⇡jfor some fixedj, then we can apply Theorem11tof=f12+· · ·+fm2, which then directly implies Theorem3.
Proof. Without loss of generality, we can assume thatf(0)<0. SinceHn 1is path connected, it suffices to show that there is a pointp+2Hn 1(G)withf(p+) 0.
We can assume that ⇡1 =kxk2. Indeed, since Gis essential, all basic invariants have degree at least2andkxk2is a linear combination of the degree2basic invariants. Letp2VR(f)and define
K={q:⇡1(q) =⇡1(p)}, the sphere centered at the origin that containsp. The functionf attains its maximum overK in a closed setM ✓K. We claim thatM\Hn 16=?. Letp0be a point in M.
We may pass to the real orbit space S = ⇡(V) associated to G and ⇡1, . . . ,⇡n and consider the compact set K := ⇡(K) = {y 2 S : y1 = ⇡1(p)}. We can write f = F(⇡1, . . . ,⇡n) for some F 2 R[y1, . . . , yn]. In this setting, our assumption states thatF is at most linear in yk. If p0 2 V \ Hn 1, then, by (6), p0 :=⇡(p0) is in the interior of S and hence in the relative interior of K. LetL={p0+tek :t2R}be the affine line throughp0 in directionek. Restricted toL, the polynomialF has degree at most1. By Lemma10and our choice ofK, the lineLmeets@Kin two pointsp ,p+andF(p )F(p0)F(p+). This implies that⇡ 1(p+)✓M and, since@K✓@S,
equation (6) shows that⇡ 1(p+)✓Hn 1. ⇤
The assumption in Theorem 11that Gis essential is essential. For example, let G =Bn act on V =Rn⇥Rby fixing the last coordinate. A set of basic invariants is given by⇡1(x, xn+1) =xn+1
and ⇡i(x, xn+1) = s2i 2(x) for i = 2, . . . , n+ 1. Pick p 2 Rn with all coordinates positive and distinct. The variety
(7) X = {(x, xn+1)2V :s2i(x) =s2i(p)fori= 1, . . . , n}
is defined over R[⇡2, . . . ,⇡n+1], but is a collection of affine lines that does not meet the reflection arrangement.
We give two further applications of Theorem11.
Corollary 12. Let G be an essential reflection group and let J ⇢ [n] with |J| = n 1. For polynomials f, f1, . . . , fm2R[⇡i:i2J], the following hold:
(i) S ={p:f1(p) 0, . . . , fm(p) 0}is nonempty if and only ifS\Hn 1(G)6=?.
(ii) f(q) 0for allq2Sif and only iff(q) 0for allq2S\Hn 1(G).
Proof. Forq2S, it suffices to prove the claim for
X := {p2V :⇡j(p) =⇡j(q)forj2J} ✓ S.
Claim (i) now follows from Theorem11. As for (ii), assume thatq2S\ Hn 1andf(q)<0. Then the same argument applied toX\{p:f(p) =f(q)}finishes the proof. ⇤ Iff 2R[V]Ghas degreedeg(f)<2 deg(⇡n), then the algebraic independence of the basic invariants implies that Theorem 11 can be applied to proof the following corollary. Under the assumption that f is homogeneous, the second part of the corollary recovers the main result of Acevedo and Velasco [1].
Corollary 13. Letf2R[V]G withdeg(f)<2dn(G) = 2 deg(⇡n). ThenVR(f)6=?if and only if VR(f)\Hn 16=?. In particular,f 0onV if and only iff 0onHn 1.
The bound on the degree is tight: For a pointp2V \ Hn 1(G), the set of solutions to f(x) :=
Xn i=1
(⇡i(x) ⇡i(p))2 = 0
is exactlyGp, which does not meetHn 1(G). The defining polynomialf(x) is of degree exactly 2 deg(⇡n). Theorem11also allows us to prove Theorem2for groups of low rank.
Proof of Theorem2forrank(G)3. Fork = rank(G), there is nothing to prove. For k = 1, we observe thatX1(p)is the sphere throughp, which meets the arrangementH1(G) of lines through the origin. Thus, the only nontrivial case isrank(G) = 3andk= rank(G) 1 = 2. This is covered
by Theorem3. ⇤
LetGbe an essential reflection group of rank 4. SinceGacts onV by orthogonal transformations, we have that⇡1(x) =kxk2andXk(p)is a subvariety of a sphere centered at the origin. Since the basic invariants are homogeneous, we may assume that⇡1(p) = 1and henceXk(p)✓Sn 1={x2 V :kxk= 1}. To prove Theorem2fork= 2we can proceed as follows. Let min and max be the minimum and maximum of⇡2overSn 1. Then it suffices to find pointspmin,pmax2H2(G)\Sn 1 with⇡2(pmin) = min and⇡2(pmax) = max. Indeed, sinceH2(G) is connected (forrank(G) 3), this shows that⇡J(V) =⇡J(H2(G))forJ={1,2}, which, by Proposition8, then proves the claim.
For the groupF4, we can implement this strategy.
Proof of Theorem2forF4. Since F4is of rank4, we only need to consider the casek= 2and can use the strategy outlined above. Let min and max be the minimum and maximum of⇡2overS3. An explicit description of⇡2forF4is
⇡2(x) = X
1i<j4
(xi+xj)6+ (xi xj)6;
see, for example, Mehta [13] or [12, Table 5]. The pointsp= (1,0,0,0)andp0= (p1
2,p1
2,0,0)are contained inH1(F4)✓H2(F4)and takes values ⇡2(p) = 1and⇡2(p0) = 32. We claim, that these values are exactly min and max, respectively.
Note that⇡2(x) =g(x21, x22, x23, x24)for
g(y) = 5s1(y)·s2(y) 4s3(y).
Let 3 ={x2R4 :x1, . . . , x4 0, x1+· · ·+x4= 1}be the standard 3-simplex. We have that
⇢(S3) = 3where⇢(x1, . . . , x4) := (x21, . . . , x24). Hence,
max = max{g(p) :p2 3} and min = min{g(p) :p2 3}.
Now,D4is a subgroup ofF4and⇡22R[s2, s4, s6]and does not depend one4(x). By Theorem2 forD4, the varietiesS3\{⇡2(x) = min}and S3\{⇡2(x) = max}both meet H3(D4). Hence, it suffices to minimize or maximizeg(x)over
3\{x2R4:x1=x2}.
This leaves us with the (standard) task to maximize and minimize a bivariate polynomialg0(s, t)of degree3over a triangle. In the plane, the polynomial has3critical points with values1,119,119. On the boundary, the extreme values are attained at the points given above. ⇤ For the rank-4reflection groupH4, the invariant⇡2(x)is a polynomial of degree12in four variables;
see, for example, [12, Table 6]. Since(B1)4 is a reflection subgroup ofF4, ⇡2 is a polynomial in the squaresx21, . . . , x24and, following the argument in the proof above, we are left with minimizing and maximizing a degree-6 polynomial g(x) over the simplex 3. However, finding the critical points is not easy and an extra computational challenge is the fact thatg(x) is a polynomial with coefficients in Q(p
5). GloptiPoly[10]numerically computes min = 165 and max = 1. These values are attained at pmin = p12(1,1,0,0) and pmax = (1,0,0,0), respectively, and both points lie inH2((B1)4)✓H2(H4). This is strong evidence for the validity of Conjecture1forH4 but, of course, not a rigorous proof.
4. Strata of higher codimension
We have seen in the previous section that every nonempty(n 1)-sparse variety meets the hyperplane arrangementHn 1. In this section want to extend this result tok-sparse varieties fork < n 1. This case is considerably more difficult but we can make good use of the techniques and ideas developed in Section3.
Let G be an essential finite reflection group acting onV ⇠=Rn. Consider aG-invariant k-sparse varietyX withk < n. IfX is nonempty, then Theorem11yields that for some reflection hyperplane H 2Hthe varietyX0:=X\H is nonempty. An inductive argument could now replaceGby some other reflection subgroupG0✓Gthat fixesH. IfX0remains sparse with respect toG0we can again apply Theorem11to obtain a pointp2Hn 2(G0)✓Hn 2(G). However, the results obtained using this strategy are far from optimal. We will briefly illustrate this forG=Sn: LetX be a nonempty k-sparseS-invariant variety for k < n. The largest subgroup of Sn that fixes a given reflection hyperplaneH2HisG0⇠=Sn 2⇥S2. Hence Theorem11only applies forX0=X\H and G0 if k=dk(G)< dn(G0) =n 2, in other words if the original varietyX is(k 3)-sparse. Inductively, this yields that every nonemptyk-sparseSn-invariant variety meetsHlwherel=bn+k2 c. However, applying the above method to the exceptional types gives nontrivial bounds.
Proposition 14. Let 6n 8. Then every nonempty 2-sparse En-invariant variety intersects Hn 2(En).
Proof. We exemplify the argument for the casen= 8. LetX be a nonempty2-sparseE8-invariant variety. By Theorem11we find a pointp2X\H7(E8). The orbit ofpmeets every hyperplane in H(E8)(see [11, Sect. 2.10]) and hence we may assume thatplies on the hyperplaneH ={x2R8: x1=x2}. Consider the subgroup G0 ⇠=D6⇢E8 acting essentially on the coordinatesx3, . . . , x8. Sinced6(D6) = 10>8 =d2(E8), we can apply Theorem11to finish the proof. ⇤ By restricting the class of invariant polynomials, we obtain better bounds than those in Proposi- tion14. In the following, a pointpis calledG-generalif it does not lie on any reflection hyperplane ofG, and hence|Gp|=|G|.
Definition 15. For a positive integer d, let G(d) be the largest number ` such that for every p 2 H`+1 there is a reflection subgroup G0 ✓ G such that p is G0-general and 2dn(G0) > d.
Moreover, we define G(k) := G(2dk(G)). That is, G(k) is the largest0`d such that for everyp2H`+1, there is a reflection subgroupG0✓Gsuch thatpisG0-general anddn(G0)> dk(G).
We call an invariant polynomial f 2 R[V]G G-finite if either VR(f) = ? or if there is a point p 2 VR(f) such that f has finitely many extreme points restricted to the sphere K = {q 2 V : kqk=kpk}.
Theorem 16. Letf2R[V]Gbe aG-finite polynomial andX = VR(f). Iff 2R[⇡1, . . . ,⇡k], then X 6= ? if and only if X\H G(k) 6= ?.
Ifd= deg(f), then
X 6= ? if and only if X\HG(d) 6= ?.
Proof. We only give a proof for the second result. The proof of the first is analogous. Suppose VR(f)6=?. We may assume thatf(0)0and, sinceHG(d) is connected, it suffices to show that there is some pointp+2HG(d) withf(p+) 0.
By assumption, there is a zero p0 2 VR(f) such that f has only finitely many extreme points restricted to K = {q : kqk = kp0k}. Let p+ 2 K be a point maximizingf over K and hence f(p+) f(p0) 0. We claim thatp+2HG(d). Otherwise, there is a reflection subgroupG0⇢G such thatp+62Hn 1(G0)and2dn(G0)> d. Let⇡10, . . . ,⇡n0 be a choice of basic invariants ofG0and, without loss of generality,⇡01(x) =kxk2. Thus,p+=⇡0(p+)is in the interior ofS=⇡0(V). We can writef =F(⇡10, . . . ,⇡n) for someF 2R[y1, . . . , yn]. On the level of orbit spaces, our assumption states that restricted toK =⇡(L) = {y2 S :y1 =⇡1(p+)}, the polynomial F has only finitely many extreme points. However,F is linear inynand thusp+is a maximum only ifp+2@K✓@S.
This is a contradiction. ⇤
5. Bounds from parabolic subgroups
The numbers G(d) and G(k) defined in Section 4 are difficult to compute in general. In this section, we compute upper bounds on these numbers coming from parabolic subgroups. LetGbe a finite irreducible reflection group. A fundamental domain ⇢V, as defined in Section 3, is a simplicial cone of dimensionn= dimV. LetH1, . . . , Hn2H(G)be the reflection hyperplanes that are facet-defining for and let ={s1, . . . , sn}⇢Gbe the corresponding reflections. Fori6=, we denote bym(i, j)the order of the cyclic group generated bysisj. TheDynkin diagramDofGis the labelled graph with vertex set{1, . . . , n}and edgesij wheneverm(i, j) 3and edge labelling m(i, j); see [11, Sec. 2.1] for details. A subgroup ofGisparabolicif it is conjugate to a subgroup generated by a subset of the reflections in ; cf. [11, Sec. 1.10].
Lemma 17. Fix a finite irreducible reflection groupGwith Dynkin diagramD. LetD0⇢D be a subdiagram obtained by removing a node fromDand letH 2H(G)be a reflection hyperplane. Then there is a parabolic subgroupW ⇢Gwith Dynkin diagramD0 andH is not a reflection hyperplane of W.
Proof. LetW be a parabolic subgroup with Dynkin diagram D0. Since every parabolic subgroup with Dynkin diagramD0is conjugate toW, it suffices to show that for everyH2H(G)there is a g2Gsuch thatgH62H(W).
Unless Gis of typeF4,Bn orI2(2m) form > 1, the hyperplanes inH(G) form a single G-orbit (see [11, Sec. 2.9, Sec. 2.10]). Hence, the claim follows, sinceH(W)(H(G). For F4, the possible proper parabolic subgroups areB3 andA1⇥A2and the result follows by inspection. ForI2(2m), there are two orbits with each2m 4roots whereas the only nontrivial proper parabolic subgroup isA1. ForBn, this follows from counting the number of elements in each of the two orbits. ⇤ The lemma yields the following result about finite reflection groups that might be interesting in its own right.
Proposition 18. LetG be a finite irreducible reflection group with Dynkin diagram D acting on a real vector space V. For k 1, letp2Hk\ Hk 1andD0⇢D be a connected subdiagram onk nodes. Then there is a parabolic subgroupW ⇢Gwith Dynkin diagramD0such thatpisW-general.
Proof. We argue by induction ons= dimV k. Fors= 0,p2V \ HdimV 1andpis by definition G-general. Otherwise, letD1⇢Dbe a subdiagram obtained by removing a leaf such thatD0✓D1
and let H1 be a reflection hyperplane of G containing p. We may use Lemma 17 to obtain a parabolic subgroupW1with Dynkin diagramD1and not containingH1as a reflection hyperplane.
In particular,pis contained in preciselys 1linearly independent reflection hyperplanes ofW1. By induction, there is a parabolic subgroupW ✓W1with Dynkin diagramD0for whichpisW-general.
In particular,W is a parabolic subgroup ofGwhich concludes the proof. ⇤ ForI✓ , letWI be the parabolic subgroup generated by the reflectionsI. We define
˜G(d) := min{|I| 1 :I✓ ,2dn(WI)> d},
and, analogously, we define˜G(k) := ˜G(2dk(G)). SinceGacts transitively on the set of fundamental domains, these definitions do not depend on the choice of . Proposition 18implies the following bound on G.
Corollary 19. G(d)˜G(d),for alld 0.
The clear advantage is a simple way to compute upper bounds on G(d) from the knowledge of parabolic subgroups of reflection groups; cf. [11]. The explicit values are given in Table1. However,
G d ˜G(d) W dn(W) k ˜G(k)
An 1/Sn 0—2n 1 bd/2c Abd/2c bd/2c+ 1 0—n 1 k
Bn 0—4n 1 bd/4c Bbd/4c+1 2(bd/4c+ 1) 0—n 1 k
Dn 0—4n 5 bd/4c+ 1 Dbd/4c+2 2(bd/4c+ 1) 0—bn2c k+ 1
bn2c+ 1—n k
I2(m) 1—2m 1 1 I2(m) m 1 1
E6 1—5 1 A2 3 1 1
6—7 2 A3 4 2 3
8—11 3 D4 6 3 4
12—15 4 D5 8 4 5
16—23 5 E6 12 5 5
E7 1—5 1 A2 3 1 1
6—7 2 A3 4 2 4
8—11 3 D4 6 3 5
12—15 4 D5 8 4 5
16—23 5 E6 12 5 6
24—35 6 E7 18 6 6
E8 1—5 1 A2 3 1 1
6—7 2 A3 4 2 5
8—11 3 D4 6 3 6
12—15 4 D5 8 4 6
16—23 5 E6 12 5 7
24—35 6 E7 18 6 7
36—59 7 E8 30 7 7
F4 1—7 1 B2 4 1 1
8—11 2 B3 6 2 3
12—23 3 F4 12 3 3
H3 1—9 1 I2(5) 5 1 1
10—19 2 H3 10 2 2
H4 1—9 1 I2(m) 5 1 1
10—19 2 H3 10 2 3
20—59 3 H4 30 3 3
Table 1. Computation for˜G(d)and˜G(k).a—brefers to the rangea, a+1, ..., b.
The columnW gives the parabolic subgroup that attains˜G.
not every reflection subgroup is parabolic (e.g. Dn ✓ Bn, I2(m) ✓ I2(2m)). Nevertheless, we conjecture that G(d)is attained at a parabolic subgroup.
Conjecture 2. For any finite reflection group G, G(d) = ˜G(d) for alld.
6. Adjoint representations of Lie groups
In this last section, we extend some of our results to polynomials invariant under the action of a Lie group. More precisely, we consider the case of a real simple Lie groupGwith the adjoint action on its Lie algebra g. We illustrate our results for the caseG= SLn. Its Lie algebrasln is the vector space of realn-by-nmatrices of trace0. The adjoint action ofSLnonslnis by conjugation:g2SLn
acts onA2slnbyg·A:=gAg 1. The following description of its ring of invariants is well-known.
We briefly recall the standard proof which immediately suggests a connection to our treatment of reflection groups; cf. [7, Ch. 12.5.3].
Theorem 20. Forn 1, R[sln]G=R[s2, . . . , sn],where sk(A) = tr(Ak) for k= 2, . . . , n. More- over,s2, . . . , snare algebraically independent.
Proof. We writeD⇢slnfor the set of diagonalizable matrices and we denote by (A) = ( 1, . . . , n) the eigenvalues ofA2D. Then for anyA2D
sk(A) = sk( (A)) = 1(A)k+ 2(A)k+· · ·+ n(A)k
and s2, . . . , sn are simply the power sums restricted to the linear subspace ⇢ D of diagonal matrices. This shows that s2, . . . , snare algebraically independent. Now for a polynomialf(X)2 R[sln]invariant under the action of SLn, the restriction to ⇠=Rn 1 is a polynomial f(x) that
is invariant under An 1. Hence f(x) = F(s2(x), . . . , sn(x)) for some F 2 R[y2, . . . , yn]. The polynomialf(X) =˜ F(s2(X), . . . , sn(X))is invariant underSLnand agrees withf onD. SinceD
contains a nonempty open set,f = ˜f as required. ⇤
In general, let T ✓G be a maximal torus with Lie algebrat✓g. If N ✓G is the normalizer of T inG, then W =N/T is a reflection group, the Weyl group, that acts on t. By Chevalley’s Restriction Theorem (see [8, Lem. 7]), the restrictionR[g] ! R[t] extends to an isomorphism of invariant rings. This yields thatR[g]Gis generated by homogeneous and algebraically independent polynomials ⇡1, . . . ,⇡m whose restriction to tgive a set of basic invariants for W. We call a G- invariant varietyX ✓gk-sparse ifX = VR(f1, . . . , fm) for somef1, . . . , fm 2R[⇡1, . . . ,⇡k]. The discriminant locusD ⇢gof Gis the Zariski closure of the orbit of the reflection arrangement H(W) ⇢ t underG. By the result of Steinberg [17] and the Restriction theorem, this is a real G-invariant hypersurface given by the vanishing of a single polynomial, called thediscriminantof G. Since G·tis dense in g,Dis the closure of the set of points with nontrivial stabilizer; see [9, Ch. 6]. This yields a stratification ofgby defining Di to be the closure of the orbit ofHi, which corresponds to the points for which the discriminant vanishes up to ordern i. Hence, the results from the previous sections generalize to Lie groups.
For the special orthogonal groupSOn, its Lie algebrason⇢slnis the vector space of skew-symmetric n-by-n matrices on which SOn acts by conjugation. Ifn = 2k+ 1, then the corresponding Weyl group is Bk and Dk ifn = 2k. Hence R[so2k+1]SO2k+1 is generated by s2(X), s4(X), . . . , s2k(X).
For n = 2k, a minimal generating set is given by s2(X), s4(X), ..., s2n 2(X) and the Pfaffian pf(X) =p
detX. Theorem2yields the following.
Theorem 21. LetG2{SLk,SOk:k2Z 1}and letX✓g beG-invariant andk-sparse. IfX is nonempty it intersectsDk.
For suitable Lie groupsG, Theorem21gives a first relation between real varieties invariant under the action ofGand the discriminant locus and Conjecture1is reasonable for this setting. It would be very interesting to explore this connection further.
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Fachbereich Mathematik und Informatik, Freie Universität Berlin, Germany E-mail address:[email protected]
Aalto Science Institute, PO Box 11000, FI-00076 Aalto E-mail address:[email protected]
Institut für Mathematik, Goethe-Universität Frankfurt, Germany E-mail address:[email protected]
