Governing singularities of symmetric orbit closures

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by  Alexander Woo

GOVERNING SINGULARITIES OF SYMMETRIC ORBIT CLOSURES
ALEXANDERWOO,BENJAMINJ.WYSER,ANDALEXANDERYONG
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ABSTRACT. We developintervalpatternavoidanceand Mars-Springer idealstostudy singu-
1
laritiesof symmetric orbitclosuresinaflagvariety. Thispaperfocusesonthe caseofthe
0
2 LevisubgroupGLp×GLq actingontheclassicalflagvariety. Weprovethatallreasonable
singularitypropertiescanbeclassifiedintermsofintervalpatternsofclans.
n
a
J

0 1. INTRODUCTION
1

1.1. Overview. Let G/B be a generalized flag variety where G is a complex, reductive
]
G algebraic group and B is a choice of Borel subgroup. A subgroup K of G is symmetric
A if K = Gθ is the fixed point subgroup for an involutive automorphism θ of G. Such a
h. subgroupisspherical,whichmeansthatitsactiononG/B bylefttranslationshasfinitely
t manyorbits.
a
m
The study of orbits of a symmetric subgroup on the flag variety was initiated in work
[
ofG.Lusztig–D.Vogan[5,14],whorelatedthesingularitiesoftheirclosurestocharacters

1 of particular infinite-dimensional representations of a certain real form G of G. Since
R
v
then, there has been a stream of results on the combinatorics and geometry of these orbit
4
7 closures. Notably, R. W. Richardson–T. A. Springer [12] gave a description of the partial
7 ordergivenbyinclusionsoforbit closures, andM.Brion [3]studiedgeneralpropertiesof
2
theirsingularities,showingthat,inmanycases,includingtheoneaddressedinthispaper,
0
. all these orbit closures are normal and Cohen-Macaulay with rational singularities. One
1
0 might also hope that the study of singularities on closures of symmetric subgroup orbits
7
wouldleadtobetterunderstandingofthegeneralrelationshipbetweenthecombinatorics
1
: associated to spherical varieties andsingularities oforbit closures on them; N.Perrin has
v
written a survey ofthis topic [11].
i
X
We initiate a combinatorial approach, backed by explicit commutative algebra compu-
r
a tations, to the study of the singularities of these orbit closures. This paper considers the
caseofthesymmetricsubgroupK = GL ×GL ofblockdiagonalmatricesinGL ,where
p q n
n = p+q. Here θ isdefined by
(1) θ(M) = I MI ,
p,q p,q
where I is the diagonal ±1 matrix with p many 1’s followed by q many −1’s on the
p,q
diagonal. In thiscase, G = U(p,q), the indefiniteunitary group ofsignature (p,q).
R
For this symmetric subgroup K, the finitely many K-orbits O and their closures Y
γ γ
can be parameterized by (p,q)-clans γ [7, Theorem 4.1] (see also [22, Theorem 2.2.8]).
Theseclansarepartialmatchingsofvertices{1,2,...,n}whereeachunmatchedvertexis
assigned a sign of + or −; the difference in the number of +’s and −’s must be p−q. We
represent clans by length n strings in N∪{+,−}, with pairs of equal numbers indicating
Date:January12,2017.
1
a matching. Let Clans denote the set of all such clans. For example, three clans from
p,q
Clans are represented bythe strings:
7,3
1+++2+1−2+, +1−++1+2+2, and ++−+++−++−.
Note that strings that differ by a permutation of the natural numbers, such as 1212 and
2121,represent the sameclan.
OneinspirationforthisworkisW.M.McGovern’scharacterization ofsingularK-orbit
closures in terms of pattern avoidance of clans [9]. Suppose γ ∈ Clans and θ ∈ Clans .
p,q r,s
Then θ = θ ...θ issaid to (pattern)include γ = γ ...γ ifthere are indicesi < i <
1 r+s 1 p+q 1 2
··· < i such that:
p+q
(1) ifγ = ±then θ = γ ; and
j ij j
(2) ifγ = γ then θ = θ .
k ℓ i i
k ℓ
Forexample,theclanγ = 1++−−1 containsthepatternθ = 1++−1, taking(i ,...,i )
1 5
to beeither (1,2,3,4,6)or (1,2,3,5,6).
Say that θ (pattern) avoids γ if θ does not include γ. The main theorem of [9] asserts
that Y issmooth ifand onlyifγ avoids the patterns
γ
1+−1, 1−+1, 1212, 1+221, 1−221, 122+1, 122−1, 122331.
On the other hand, in [16, Section 3.3] it is noted that Y is non-Gorenstein, while
1++−1
Y is Gorenstein, even though 1+ +− −1 pattern contains 1 + +− 1. Therefore, a
1++−−1
more general notion will sometimes be required to characterize which K-orbits satisfy a
particular singularity property.
Suppose that P is anysingularity mildnessproperty, bywhich wemean a local prop-
ertyofvarietiesthatholdsonopensubsetsandisstableundersmoothmorphisms. Many
singularity properties, such as being Gorenstein, being a local complete intersection (lci),
beingfactorial, having Cohen-Macaulayrank ≤ k, or having Hilbert-Samuel multiplicity
≤ k, satisfy these conditions. For such a P, consider two related problems:
(I) Which K-orbit closures Y are globally P?
γ
(II) Whatisthe non-P-locus of Y ?
γ
This paper gives a universal combinatorial language, interval pattern avoidance of
clans, to answer these questions for any singularity mildness property, at the cost of po-
tentiallyrequiring aninfinite numberofpatterns. Thislanguage isalsousefulfor collect-
ing and analyzing data and partial results. We present explicit equations for computing
whetheraproperty holdsataspecificorbitO onanorbitclosure Y . Notethatallpoints
α γ
ofO ⊆ Y are locallyisomorphic to one another, since the K-action can beused to move
α γ
any point of O isomorphically to any other point. Since the non-P-locus is closed, it is
α
a union of K-orbit closures. Consequently, for any given clan γ, (II) can be answered by
findinga finite setofclans{α},namelythose indexingthe irreducible components ofthe
non-P-locus. (I)asksifthis setis nonempty.
This situation parallels that for Schubert varieties. In that setting, the first and third
authorsintroducedintervalpatternavoidanceforpermutations,showingthatitprovides
acommonperspectivetostudyallreasonablesingularitymeasures[17,18,15]. Thispaper
gives the first analogue ofthose results for K-orbit closures.
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Some properties P hold globally on every Y . For those cases, the above questions are
γ
unnecessary. Forexample,thisistrue whenP =“normal”andP =“Cohen-Macaulay”in
the case (G,K) = (GL ,GL ×GL ) of this paper. (This is not the case for all symmetric
n p q
pairs(G,K).)
As is explained in [9], the above answer to (I) for P =“smooth” also is the answer
for the property P =“rationally smooth”. (Recall rational smoothness means that the
local intersection cohomology of Y (at a point of O ) is trivial.) However, the answer to
γ α
question(II),whichasksforacombinatorialdescriptionofthe(rationally)singularlocus,
isunsolvedexceptinsomespecialcases[16]. Actually,itisunknownwhetherthesingular
locus andrationally singularlocus coincide for all orbit closures (butsee Conjecture 7.5).
For most finer singularity mildness properties, answers to both (I) and (II) are un-
known. Perhaps the most famous such property comes from the Kazhdan-Lusztig-Vogan
(KLV) polynomials P (q). These polynomials are the link to representation theory that
γ,α
originally motivated the study of K-orbit closures. For (G,K) = (GL ,GL × GL ), the
n p q
KLV polynomial P (q) ∈ Z [q] is the Poincare´ polynomial for the local intersection co-
γ,α ≥0
homology of Y at any point of the orbit O [5]. Rational smoothness of Y along O is
γ α γ α
henceequivalenttotheequalityP (q) = 1. Moregenerally,foranyfixedk > 1,theprop-
γ,α
erty P =“P (1) ≤ k” behavesas a singularity mildnessproperty on K-orbit closures by
γ,α
recent work of W. M. McGovern [10], but his proof uses representation theory, and the
algebraicgeometry isnot well-understood.
1.2. Main ideas. Each K-orbit closure Y is a union of K-orbits O ; let ≤ denote the
γ α
Bruhat (closure) order on K-orbit closures on GL /B. (This means α ≤ γ if and only if
n
Y ⊆ Y ). Also, let [α,γ] and [β,θ] be intervals in Bruhat order on Clans and Clans ,
α γ p,q r,s
respectively.
Define[β,θ]to intervalpatterncontain [α,γ],and write [α,γ] ֒→ [β,θ]if:
(a) there are indices
I : i < i < ··· < i
1 2 p+q
which commonly witnessthe containment of γ intoθ and α intoβ,
(b) θ and β agree outside of these indices;and
(c) ℓ(θ)−ℓ(β) = ℓ(γ)−ℓ(α),
where ifγ = γ γ ...γ then
1 2 p+q
ℓ(γ) := j −i−#{γ = γ : s < i < t < j}.
s t
γi=Xγj∈N
Notice β is determined by α,γ, θ, and the set of indicesI. In particular, β is the unique
clan Φ(α) that agrees with α on I and agrees with θ on {1,...,n} \ I. Thus, we define a
clan θ to interval pattern contain [α,γ] if [α,γ] is contained in [Φ(α),θ]. Similarly, we can
speakofθ avoidinga listof intervals.
Example 1.1. Let [α,γ] = [+ − −+,1212] and [β,θ] = [1 + − − +1,123231]. Then one can
check[β,θ]interval contains [α,γ],using the middlefour positions. (cid:3)
Example1.2. Let[α,γ] = [+−,11] and[β,θ] = [++−,1+1]. If[β,θ]contains [α,γ]itmust
be using the underlined positions. However, ℓ(γ)−ℓ(α) = 1 while ℓ(θ)−ℓ(β) = 2. Thus
[β,θ]doesnotcontain [α,γ]. (cid:3)
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Define
C := {[α,γ] : α ≤ γ in some Clans } ⊆ Clans×Clans
p,q
where
Clans = Clans .
p,q
p,q
[
Declare (cid:22) tobe the poset relation on C generated by
C
• [α,γ] (cid:22) [β,θ]if[β,θ]interval pattern contains [α,γ],and
C
• [α,γ] (cid:22) [α′,γ]ifα′ ≤ α.
C
For any poset (S,(cid:22)), an upper order ideal is a subset I of S having the property that
wheneverx ∈ I,wealsohavey ∈ I forally ≥ x. Thefollowingtheoremprovidesabasic
language to express answersto (I)and (II).
Theorem 1.3. LetP be asingularitymildnessproperty(seeDefinition2.1).
(I) Thesetofintervals[α,γ] ∈ C suchthatP failsoneachpointofO ⊆ Y isanupperorder
α γ
idealin(C,(cid:22) ).
C
(II) Thesetof clansγ suchthatP holdsatallpoints ofY arethosethatavoidallthe intervals
γ
[α ,γ ]constituting some(possiblyinfinite)setA ⊆ C.
i i P
Forsimplicity, weworkoverC,butourresults arevalidoveranyfieldofcharacteristic
not 2,with the caveatthat evenfor the same P, the set A maybe field dependent.
P
Although stated in combinatorial language, as we will see, Theorem 1.3 follows from
a geometric result, Theorem 4.5, which establishes a local isomorphism between certain
“slices” of the orbit closures. Rather than working directly with the K-orbit closures on
G/B, it is easier to establish this isomorphism using particular slices of B-orbit closures
on G/K given by J. G. M. Marsand T.A. Springer[6].
ThespaceG/K = GL /(GL ×GL )istheconfiguration spaceofall splittings ofCn as
n p q
adirectsumV ⊕V ofsubspaces,withdim(V ) = panddim(V ) = q. Indeed,G = GL (C)
1 2 1 2 n
actstransitivelyonthesetofallsuchsplittings,andK = GL ×GL isthestabilizerofthe
p q
“standard” splitting of Cn as he ,...,e i ⊕he ,...,e i, where e ,...,e is the standard
1 p p+1 n 1 n
basis of Cn. A point gK/K ∈ G/K, with g ∈ GL (C) an invertible n × n matrix, is
n
identified with the splitting whose p-dimensional component is the span of the first p
columns ofg and whose q-dimensional component isthe span of the lastq columns ofg.
B-orbitsonG/K areinbijectionwithK-orbitsonG/B(bothorbitsetsbeinginbijection
with the B ×K-orbits on G). In addition, this bijection preserves all mildnessproperties.
This is because the orbit closures correspond via the two locally trivial fibrations G →
G/K and G → G/B, each of which hassmooth fiber. The B-orbit on G/K corresponding
to the clan γ will be denoted by Q , and its closure will be denoted by W . For any
γ γ
singularity mildnessproperty P, this discussion impliesthe following:
Observation 1.4. Y is P along O ifand only ifW is P along Q .
γ α γ α
In [21], the second and third authors considered certain open affine subsets of K-orbit
closures on G/B which they called patches. That work also introduced patch ideals of
equations which set-theoretically cut out the patches. Experimental computation using
patch idealsledto several of the conjectures which appearin [21, 16].
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In contrast, the Mars-Springer slices are not open affine pieces of the orbit closures.
However, as we will see, they are essentially as good, in that they carry all of the local
information that we are interested in. They are analogues of Kazhdan-Lusztig varieties,
which playa similarrole in the study ofSchubert varieties.
In this paper we introduce Mars-Springer ideals (see Section 4) with explicit equations
that set-theoretically cut out the Mars-Springer slices. These equations are conjectured to
alsobe scheme-theoretically correct; seeConjecture 7.1. These are the K-orbit versions of
Kazhdan-Lusztig ideals, which define the aforementioned Kazhdan-Lusztig varieties. The
latter ideals have been of use in both computational and theoretical analysis of Schubert
varieties (see, for example, [19, 13] and the references therein). In the same vein, we
mention a practical advantage of the Mars-Springer ideal over the patch ideal. Gro¨bner
basiscalculationswiththeformerareseveraltimesfasterthanthoseofthelatter,asfewer
variablesare involved.
1.3. Organization. In Section 2, we present some preliminaries. In particular, we more
preciselydefine“mildnessproperties,”givingexamplesandestablishingsomebasicfacts
that we will need. We next recall the attractive slices of Mars-Springer [6]. Finally, we
describe the quasi-projective variety structure of G/K for the case (G,K) = (GL ,GL ×
n p
GL ) of this paper. In Section 3, we give explicit affine coordinates for the Mars-Springer
q
slice(Theorem 4.3). Section 4definesthe Mars-Springervarietyanditsideal. Itculminates
with Theorem 4.5, which asserts that certain Mars-Springer varieties are isomorphic to
others. ThistheoremisthekeytoourproofofTheorem1.3. InordertoproveTheorem4.5,
wedevelopcombinatoricsofintervalembeddingsusingearlierworkofthesecondauthor
[20]ontheBruhatorderonclans;thisisSection5. WethengiveourproofsofTheorem4.5
and Theorem 1.3in Section 6. Weconclude with problems andconjectures in Section 7.
2. PRELIMINARIES
2.1. Singularity Mildness Properties. We define the class of local properties that we are
interested in,with specificexamples.
Definition2.1. SupposethatP isalocalpropertyofalgebraicvarieties,meaningthatP isverified
at a point w of an algebraic variety W based solely on the local ring O . We say that P is a
W,w
singularity mildness property (or simplyamildness property) ifP is
(1) Open,meaning thatthe P-locus of any variety isopen; and
(2) Stable under smooth morphisms. Precisely, we mean that if X and Y are any varieties,
and f : X → Y is any smooth morphismbetween them, then for any x ∈ X, X is P at x
ifand only ifY is P atf(x).
For any smooth variety S and for any variety X, the projection map S × X → X is a
smooth morphism. Thus:
Observation 2.2. If P is a mildness property, S is a smooth variety, and X is any variety, then
for any x ∈ X and for any s ∈ S,X isP atx ifand only if S ×X isP at(s,x).
Thenext lemmagives a(non-exhaustive) list ofexamplesofmildnessproperties P.
Lemma2.3. Examplesof mildnesspropertiesP include:
(i) Reducedness;
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(ii) Normality;
(iii) Smoothness;
(iv) lci-ness;
(v) Gorensteinness.
Proof. That these properties are open (on varieties) is well-known. That they are stable
undersmoothmorphismsfollowsfromresultsof[8,§23]. Indeed,letX andY bevarieties,
withf : X → Y asmooth morphism betweenthem. Letx ∈ X begiven,andlety = f(x).
Now, f is flat, and also the fiber F over y is smooth over the residue field κ(y), hence
y
regular. Thus O is regular, hence also lci and Gorenstein. Now (iii) and (v) follow
Fy,x
from Theorems 23.7 and 23.4 [8, §23], respectively, while (iv) is mentioned in the remark
following Theorem 23.6ofloc. cit.;asindicated there, further detailscan be found in [1].
For (i) and (ii), we appeal to [8, Corollary to Theorem 23.9]. Note that this result re-
quires that the fiber ring corresponding to any prime ideal of O be reduced (resp. nor-
Y,y
mal), rather than just the fiber ring of the maximal ideal. In our case, all such fiber rings
are once again regular (hence both reduced and normal). Indeed, if p is any prime of
O , then p corresponds to an irreducible closed subvariety Y′ of Y. The inverse image
Y,y
schemef−1(Y′)issmooth overY′,andthefiberringO ⊗ κ(p)isthegenericfiberof
X,x OY,y
f−1(Y′) → Y′. (cid:3)
2.2. Mars-Springer slices. Let W be any irreducible variety with an action of a Borel
group B, letw ∈ W be given, andlet Q = B ·w. The following definition isfrom [6]:
Definition 2.4. Anattractiveslice toQatw inW isalocallyclosedsubsetS of W containing
w such that:
(1) Therestrictionofthe actionmap B ×S → W isasmooth morphism;
(2) dim(S)+dim(Q) = dim(W);and
(3) Thereexistsamap λ : G → B suchthat
m
(a) S isstable under Im(λ);
(b) w is afixedpointof Im(λ);and
(c) The action map G × S → S extends to a morphism A1 × S → S which sends
m
{0}×S tow.
The following result justifies our use of slices in studying singularities, since taking
slicespreserves the local properties that we are interested in.
Lemma 2.5. Given notation as in Definition 2.4 and a mildness property P, the variety W is P
atw ifand only if S is P atw.
Proof. Indeed, the action map B ×S → W is smooth, so W is P at w if and only if B ×S
isP at(1,w),which is the case ifandonly ifS isP atw, byObservation 2.2. (cid:3)
Let G be a complex algebraic group with an involutive automorphism θ, with K = Gθ
the corresponding symmetric subgroup. Suppose that T ⊆ B are a θ-stable maximal
torus and Borel subgroup of G, respectively. Let N be the normalizer of T in G, and let
W = N/T bethe Weyl group.
GivenaB-orbitQonG/K,wenowdefinetheMars-SpringersliceS toQataspecially
x
chosen point x ∈ Q. Letting U− be the unipotent radical of the Borel subgroup opposite
6
to B, thisslice is ofthe form
(2) S := (U− ∩ψ(U−))·x,
x
where ψ isa certain involution on G.
Tobe precise, considerthe set
V := {x ∈ G | x(θx)−1 ∈ N}.
In(2), we pickany
x := xK/K ∈ Qwith x ∈ V;
such a choice exists by [12, Theorem 1.3]. Let η be the image in W of xθ(x)−1 ∈ N. The
element xθ(x)−1 ∈ N may depend upon the choice of x, but η, its class modulo T, does
not. Thenin (2)we take
ψ := c ◦θ,
η
where c denotes conjugation by η. With these choices, it is shown in [6, Section 6.4]
η
that S , as defined in (2), is indeed an attractive slice to Q at x in G/K, in the sense of
x
Definition 2.4.
Thefollowing lemmaismore orlessformal; we includea proof for completeness:
Lemma2.6. With notation asabove, if W is any B-orbitclosureon G/K containing Q, then
(i) Thescheme-theoreticintersectionW ∩S isreduced,and
x
(ii) W ∩S is anattractive slicetoQat xinW.
x
Proof. Itis easytosee that the diagram
B ×(W ∩S ) W
x
B ×S G/K
x
is Cartesian. Therefore, since the action map along the bottom is smooth, the (restricted)
action map along the top is also smooth, since smooth morphisms are stable under base
extension. Since W is reduced and reducedness is a mildness property by Lemma 2.3,
B×(W ∩S )isreduced,andthusW ∩S isreduced,byObservation 2.2. Thisproves(i).
x x
For (ii), the preceding observation regarding the smoothness of the map along the top
oftheabove diagramverifiesDefinition 2.4(1)forS = W ∩S . Thismapfurthermore has
x
the same relative dimension asthe one along the bottom, which impliesthat
dim(G/K)−dim(W) = dim(S )−dim(W ∩S ).
x x
Since S isan attractive slice to Qin G/K, we knowthat
x
dim(S )+dim(Q) = dim(G/K),
x
and combining these twoequations gives
dim(Q)+dim(W ∩S ) = dim(W),
x
asrequired by Definition 2.4(2).
That W ∩S satisfies Definition 2.4(3) is obvious: Let λ : G → B be a one-parameter
x m
subgrouphavingproperties(a)and(b)relativetothesliceS . ThenclearlyW∩S isstable
x x
underIm(λ),xisstillafixedpointofIm(λ),andtheactionmapG ×(W ∩S ) → W ∩S
m x x
7
stillextendstoamapA1×(W ∩S ) → W ∩S sending{0}×(W ∩S )tox(simplyrestrict
x x x
(cid:3)
the original extension tothe smallersubset).
Corollary 2.7. For (G,K) = (GL ,GL × GL ), let clans α and γ be given, with α ≤ γ in
n p q
Bruhatorder. LetX denotetheMars-SpringerslicetoQ ataparticularpointx ∈ Q inG/K.
α α α α
ThenthesliceW ∩X is irreducible.
γ α
Proof. It is observed in [3] that all B-orbit closures on G/K for this particular case are
“multiplicity-free”, which impliesby [3,Theorem 5]thatW hasrational singularities. In
γ
particular, it is normal. Since normality is a mildness property by Lemma 2.3, W ∩ X
γ α
is also normal. Now, the “attractive” condition of slices (cf. part (3) of Definition 2.4)
ensures that W ∩X is connected because x must lie in every connected component of
γ α α
it. Beingboth normal and connected, W ∩X must beirreducible. (cid:3)
γ α
Remark 2.1. The result of Corollary 2.7 in fact holds for any symmetric pair (G,K) such
that all of the B-orbit closures on G/K are multiplicity-free. Other such cases include
(GL ,Sp ) and (SO ,GL ), as noted in [3]. However, not all symmetric pairs (G,K)
2n 2n 2n n
have this property. For those that do not, B-orbit closures on G/K can be non-normal,
and the Mars-Springer slices can infact be reducible. One exampleis(GL ,O ). (cid:3)
n n
2.3. GL /(GL × GL ) as a quasi-projective variety. We now discuss the structure of
p+q p q
G/K = GL /(GL × GL ) as a quasi-projective variety and its affine patches. The
p+q p q
statementsofthissection could beextracted from standard definitions andresults, butto
the best of our knowledge, this has never been made explicit in the literature, so we take
the opportunity to doso here.
Asstated intheIntroduction, G/K istheconfiguration spacewhose pointscorrespond
to splittings of Cn as a direct sum V ⊕V of subspaces, with dim(V ) = p and dim(V ) =
1 2 1 2
q. Hence it is natural to identify G/K with a subset of the product of Grasmmanians
Gr(p,Cn)×Gr(q,Cn). WecanrealizeGr(p,Cn)×Gr(q,Cn)asasubvarietyofP(np)−1×P(nq)−1
using the Plu¨cker embedding into each factor. The Segre embedding then realizes this
product asa subvariety of P(np)(nq)−1,so Gr(p,Cn)×Gr(q,Cn) isa projective variety.
Weletp andq denotethePlu¨ckercoordinatesonGr(p,Cn)andGr(q,Cn)respectively,
S T
where S andT respectively range over all subsets of{1,...,n} of sizep (respectively q).
Not every pair V and V of subspaces of Cn gives a splitting. In order for V and V
1 2 1 2
to form a direct sum, a basis for V must be linearly independent of a basis of V . Let M
1 2
be a matrix whose first p columns are a basis for V and whose last q columns are a basis
1
for V . If we take the Laplace expansion of detM using the first p columns and identify
2
determinants ofsubmatrices with Plu¨ckercoordinates, then we get
(3) detM = (−1)Sp q ,
S S
S⊂{1,...,n},#S=p
X
where S := {1,...,n} \ S and (−1)S := (−1)(Ps∈Ss)−(p+21). Hence we can identify G/K
with the open subset of Gr(p,Cn) × Gr(q,Cn) where (3) is nonzero. This gives G/K the
structure of a quasi-projective variety.
TheproductP(np)−1×P(nq)−1ofprojectivespaceshasacoveringbyaffinepatches{US,T :=
U × U }, where U is the patch where p is nonzero and U is the patch where q is
S T S S T T
nonzero. Thus Gr(p,Cn) × Gr(q,Cn) is covered by these affine patches. As explained
8
in [4, Section 9.1], any subspace V ∈ Gr(p,Cn) having Plu¨cker coordinate p (V ) 6= 0 has
1 S 1
a basis {e + a (V )e } where e denotes the i-th standard basis vector, and the
s r6∈S s,r 1 r s∈S i
functions a are local coordinates on the patch U ∩ Gr(p,Cn). Over this patch, there is
s,r S
P
an algebraic section φ of the projection map M0 → Gr(p,Cn), where M0 denotes the
S n×p n×p
n × p matrices of full rank p. The section φ sends the subspace V to the matrix whose
1
columns are preciselythe aforementioned basis.
Likewise,anysubspaceV ∈ Gr(q,Cn)havingPlu¨ckercoordinateq (V ) 6= 0hasabasis
2 T 2
{e + b (V )e } ,andtheb arelocalcoordinatesonthepatchU ∩Gr(q,Cn). Over
t r6∈T t,r 2 r t∈T t,r T
thispatch, thereisasimilarsection φ whichsendsV tothefullrankn×q matrixwhose
T 2
P
columns are thisbasis.
Taking the products of the sections φ and φ , over U ∩ (Gr(p,Cn) × Gr(q,Cn)) we
S T S,T
have an algebraic map φ from G/K into M0 × M0 , which naturally embeds into
S,T n×p n×q
M by simple concatenation of matrices. OverG/K, this map takesvalues in G = GL
n×n n
and givesan algebraicsection φ of the projection map G → G/K.
S,T
3. AFFINE COORDINATES FOR THE MARS-SPRINGER SLICE
We provide explicit coordinates for the Mars-Springer slice for the case that (G,K) =
(GL ,GL ×GL ).
n p q
3.1. The affinespace S . We definean affine space ofmatrices associated to each clan α.
α
First, letw ∈ S (inonelinenotation) bedefinedasfollows. From lefttoright, 1through
α n
p are assigned to the +’sand left endsof matchings. Assign {p+1,...,n} aswe read the
clan from left to right. When we encounter a −, we assign the smallest unused number,
and when we encounter the left end of a matching, we immediately assign the smallest
unused numberfrom {p+1,...,n} to the corresponding rightend ofthe matching.
Example3.1. Ifα = 122133 ∈ Clans then w = 125436. (cid:3)
3,3 α
Wenowconstruct, instages, the generic matrixM (z) ofS . First, for i = 1,...,n:
α α
(O.1) Ifα = ±, then row ihasa1 in position w (i).
i α
(O.2) If(i < j) isa matching ofα, then row ihas1’sin positions w (i) and w (j).
α α
(O.3) If (j < i) is a matching of α, then row i has a −1 in position w (j) and a 1 in
α
position w (i).
α
Apivotisthenorthmost1ineachcolumn. (Notethatw ischosenpreciselysothatthe
α
pivots in the first p columns, and in the last q columns, occur northwest to southeast.) In
the first p columns only, setto 0all entries
(Z.1) in the same row asapivot;
(Z.2) above a pivot (and in thatpivot’s column);
(Z.3) between the pivot 1 and the corresponding −1 of a column (for all columns to
which this applies);or
(Z.4) right of a−1.
Inthe rightmost q columns, setto 0 allentries
(Z.5) in the same row asapivot;
(Z.6) above a pivot (and in thatpivot’s column);
9
(Z.7) between the pivot 1 and the corresponding 1 below it in the same column (for all
columns to which thisapplies); or
(Z.8) right of a1 which isthe second 1 inits column.
The remaining entries z of M (z) are arbitrary, with an exception for each pair of
ij α
matchings of α which are “in the pattern 1212”. For each such pair of matchings (i < k),
(j < ℓ) with i < j < k < ℓ:
(Z.9) Position(ℓ,w (i))hasentryz (asusual)whereasits“partner”position(ℓ,w (k))
α ℓ,wα(i) α
hasentry −z .
ℓ,wα(i)
Example 3.2. Let α = 1+12−2 ∈ Clans . Then w = 124365. The 1’s and −1’s are first
3,3 α
placedinto a 6×6 matrix, followed by placementof0’sasfollows:
1 · · 1 · · 1 0 0 1 0 0
· 1 · · · · 0 1 0 0 0 0
−1 · · 1 · · −1 0 0 1 0 0
7→ .
· · 1 · 1 · 0 0 1 0 1 0
   
 · · · · · 1  · · 0 0 0 1
   
 · · −1 · 1 ·  · · −1 · 1 0
   
   
Since α has no 1212-patterns, the remaining unspecialized entries of the matrix are ar-
bitrary. Thus S is an affine space of dimension 5 (the codimension of Q in
1+12−2 1+12−2
GL /K), with its genericentry beingthe matrix
6
1 0 0 1 0 0
0 1 0 0 0 0
−1 0 0 1 0 0
M (z) = .
1+12−2 0 0 1 0 1 0
 
z z 0 0 0 1
 5,1 5,2 
z z −1 z 1 0
 6,1 6,2 6,4 
  (cid:3)
Example 3.3. Now, consider α = 1+21−2. Here w = 123465. The placement of 1’s and
α
−1’s, and then 0’s,isachieved by:
1 · · 1 · · 1 0 0 1 0 0
· 1 · · · · 0 1 0 0 0 0
 · · 1 · 1 ·  0 0 1 0 1 0
7→ .
−1 · · 1 · · −1 0 0 1 0 0
   
 · · · · · 1  · · 0 0 0 1
   
 · · −1 · 1 ·  · · −1 · 1 0
   
   
The 1212-pattern in positions 1 < 3 < 4 < 6 dictates that the unspecialized positions
(6,w (1)) = (6,1) and (6,w (4)) = (6,4) are negatives of one another. Thus S is an
α α 1+21−2
affine space ofdimension 4 (the codimension ofQ in GL /K), and
1+21−2 6
1 0 0 1 0 0
0 1 0 0 0 0
 0 0 1 0 1 0
M (z) = .
1+21−2 −1 0 0 1 0 0
 
z z 0 0 0 1
 5,1 5,2 
z z −1 −z 1 0
 6,1 6,2 6,1 
 
10