A note on the codimension of the linear section of the Lagrangian-Grassmannian L(6,12)

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by  Jesús Carrillo-Pacheco

A NOTE ON THE CODIMENSION OF THE LINEAR SECTION OF THE
LAGRANGIAN-GRASSMANNIAN L(6,12)
JESU´SCARRILLO–PACHECO,FAUSTOJARQU´IN–ZA´RATE,MAURILIOVELASCO–FUENTES,
7 ANDFELIPEZALDIVAR
1
0
2 ABSTRACT. Considera2n-dimensionalsymplecticvectorspaceEoveranarbitraryfield
F.Givenacontractionmapf :∧nE→∧n−2EsuchthattheLagrangian–Grassmannian
n
L(n,2n)=G(n,2n)∩P(kerf),where∧rEdenotesther-thexteriorpowerofEand
a
P(kerf)istheprojectivizationofkerf.Inthispaper,forasymplecticvectorspaceEof
J
dimensionn=6,weprovethatthesurjectivityofthecontractionmapf :∧6E→∧4E
8
dependsonthecharacteristicofthebasefieldandwecalculatethecodimensionofthe
2 linearsectionP(kerf)⊆P(∧6E)foranycharacteristic.

]
G
S
1. INTRODUCTION
.
h
t LetE bea2n-dimensionalsymplecticvectorspaceoveranarbitraryfieldFwithsym-
a
plecticform(cid:104), (cid:105). Considerthecontractionmapf :∧nE →∧n−2E givenby
m
(cid:88)
[ (1.1) f(w1∧···∧wn)= (cid:104)ws,wt(cid:105)w1∧···∧w(cid:98)s∧···∧w(cid:98)t∧···∧wn,

1≤s<t≤n
1
v where w(cid:98) means that the corresponding term is omitted. Our main result shows that, in
2 general, the map f is not surjective. Since, by [2] the Lagrangian-Grassmannian variety
4 L(n,2n) is cut out by the projectivization P(kerf) of the kernel of f, it follows that
2 the codimension of L(n,2n) in its Plu¨cker embedding is not C2n − C2n , where Cm
8 n n−2 n
denotes the binomial coefficient. Specifically, we prove that for n = 6, and a field of
0
. characteristic 3, the contraction map f : ∧6E → ∧4E given by (1.1) is not surjective.
1
To prove this, we use a combinatorial description in Lemma 1 of the set of indices that
0
label the Plu¨cker linear relations that is then used to describe the linear section P(kerf)
7
1 thatcutsouttheLagrangian-GrassmannianL(6,12)intheGrassmannianvarietyG(6,12)
: in any characteristic. As a consequence we show that the codimension of L(6,12) in its
v
i Plu¨ckerembeddingdependsofthecharacteristicofthebasefield.
X
Thepaperisorganizedasfollows.InSection2werecallsomeresultsofthecontraction
r map (1.1) and the Lagrangian–Grassmannian. In Section 3 we give an explicit example
a
wherethecontractionmapisnotsurjectiveandgiveallthedetailsinvolvedtoobtainthe
linearsectionP(kerf)thatdefinesL(6,12).
2. PRELIMINARIES
LetE bea2n–dimensionalvectorspaceoverFequippedwithanon-degeneratesym-
plectic form (cid:104) , (cid:105). Define the set I((cid:96),m) = {α = (α ,...,α ) : 1 ≤ α < ··· <
1 (cid:96) 1
2010MathematicsSubjectClassification. 14M15,15B75,15B99.
Keywordsandphrases. Lagrangian–Grassmannianvariety;exterioralgebra;alternatingbilinearform.
Jesu´sCarrillo-Pacheco,FaustoJarqu´ın-Za´rateandMaurilioVelasco-FuentesaresupportedbytheCryptogra-
phyLaboratoryProjectPI2013-29.Secretar´ıadeCiencia,Tecnolog´ıaeInnovacio´ndelDistritoFederal(SECITI),
Me´xico.
1
2 J.CARRILLO–PACHECO,F.JARQU´IN–ZA´RATE,M.VELASCO–FUENTES,ANDF.ZALDIVAR
α ≤ m}, such that α ∈ N, and the support of α = (α ,...,α ) ∈ I((cid:96),m) as the set
(cid:96) i 1 (cid:96)
supp(α):={α ,...,α }. Thus,allindicesαareorderedsetsof(cid:96)differentintegersinthe
1 (cid:96)
set {1,2,...,m}. In what follows, all indices α are sets with (cid:96) different elements in the
set{1,2,...,m},anduptopermutation,wemay(anddoso)thinkoftheminI((cid:96),m).
Chooseabasis{e ,...,e }ofthesymplecticspaceE suchthat
1 2n
(cid:40)
1 ifj =2n−i+1,
(cid:104)e ,e (cid:105)=
i j
0 otherwise.
Then,forα=(α ,...,α )∈I(n,2n)write
1 n
eα :=eα1∧···∧eαn,
eαst :=eα1∧···∧e(cid:98)αs∧···∧e(cid:98)αt∧···∧eαn,
p :=p ,
iαst(2n−i+1) iα1···α(cid:98)s···α(cid:98)t···αn(2n−i+1)
wheree andα meansthatthecorrespondingtermisomitted. Denoteby∧nE then-th
(cid:98)αk (cid:98)k (cid:80)
exteriorpowerofE,whichisgeneratedby{e :α∈I(n,2n)}.Forw = p e ∈
α α∈I(n,2n) α α
∧nE,thecoefficientsp arethePlu¨ckercoordinatesofw. In[2,Proposition6]thekernel
α
ofthecontractionmapf ischaracterizedasfollows: Forw = (cid:80) p e ∈ ∧nE
α∈I(n,2n) α α
writteninPlu¨ckercoordinates,wehavethat
n
(cid:88)
w ∈kerf ⇐⇒ p =0, forallα ∈I(n−2,2n).
iαst(2n−i+1) st
i=1
In [2, Section 3] these linear forms were given the following description: For α ∈
st
I(n−2,2n)definethelinearpolynomials
n
(cid:88)
Π := c X ,
αst i,αst,2n−i+1 i,αst,2n−i+1
i=1
with
(cid:40)
1 if|supp{i,α ,2n−i+1}|=n,
c = st
i,αst,2n−i+1 0 otherwise,
henceΠ arepolynomialsintheringF[X :α∈I(n,2n)].Fromthefollowingformula
αst α
inPlu¨ckercoordinates
(2.1) X +X +···+X =0,
1(cid:116)2n 2(cid:116)(2n−1) n(cid:116)(n+1)
wherethesymbols(cid:116)aretobereplacedbyelementsα ∈I(n−2,2n),weobtainhomo-
st
geneouslinearequations,thatwecallaPlu¨ckerlinearrelationsink-variables,
(2.2) Π := X +X +···+X =0
αst 1,αst,2n 2,αst,(2n−1) n,αst,(n+1)
wherethetermX doesnotappearif|supp{i,α ,(2n−i+1)}|<n. When
i,αst,(2n−i+1) st
this happens we say Π is a k-plane. For the system of homogeneous linear equations
αst
Π ,α ∈ I(n−2,2n),wedenotebyB itsassociatedmatrix. ClearlythematrixB is
αst st
oforderC2n ×C2n. Forexample,ifn=6formula(2.2)becomes
n−2 n
(2.3) X +X +X +X +X +X =0,
1(cid:116)C 2(cid:116)B 3(cid:116)A 4(cid:116)9 5(cid:116)8 6(cid:116)7
whereA=10,B =11,C =12.
Recall that a vector subspace W of E is isotropic iff for all x,y ∈ W we have that
(cid:104)x,y (cid:105)=0,andifW isisotropicitsdimensionisatmostn.TheLagrangian-Grassmannian
NOTEONTHECODIMENSIONOFTHELINEARSECTIONOFL(6,12) 3
L(n,2n)istheprojectivevarietygivenbytheisotropicvectorsubspacesW ⊆E ofmax-
imaldimensionn:
L(n,2n)={W ∈G(n,2n):W isisotropicandn-dimensional},
where G(n,2n) denotes the Grassmannian variety of vector subspaces of dimension n
of E. The Plu¨cker embedding is the regular map ρ : G(n,2n) → P(∧nE) given on
each W ∈ G(n,2n) by choosing a basis w ,...,w of W and then mapping the vector
1 n
subspace W ∈ G(n,2n) to the tensor w1∧···∧wn ∈ ∧nE. Since choosing a different
basisofW changesthetensorw1∧···∧wnbyanonzeroscalar,thistensorisawell-defined
elementintheprojectivespaceP(∧nE)(cid:39)PN−1,whereN =Cn2n =dimF(∧nE).Under
thePlu¨ckerembedding,theLagrangian-Grassmannianisgivenby
L(n,2n)={w1∧···∧wn ∈G(n,2n):(cid:104)wi,wj(cid:105)=0forall1≤i<j ≤n}.
Using the contraction map f : ∧nE → ∧n−2E given by (1.1), if P(kerf) is the pro-
jectivization of kerf, in [2] it is proved that L(n,2n) = G(n,2n)∩P(kerf). We call
P(kerf)thelinearsectionthatdefinesL(n,2n)inP(∧nE).
3. NONSURJECTIVITYOFTHECONTRACTIONMAPINchar(F)=3
The purpose of this section is twofold: First, to provide a description, completely ex-
plicitandself-containedofthelinearspaceP(kerf)forn = 6, foranyfieldF, andthen
using this characterization we give an example of the non surjectiviy of the contraction
mapforafieldofcharacteristic3.
Let P = (1,C),P = (2,B),P = (3,A),P = (4,9),P = (5,8),P = (6,7),
1 2 3 4 5 6
whereA=10,B =11,C =12,asinSection2,Σ ={P ,P ,...,P },andC (Σ )the
6 1 2 6 2 6
set of all combinations of 6 objects taken 2 ata time. For 1 ≤ α < α ≤ 12 such that
1 2
α +α (cid:54)=13,wedefinethefollowingset
1 2
Σ{α ,α }={(α ,α ,P )∈I(4,12): i+α (cid:54)=13,α +13−i(cid:54)=13,j =1,2}.
1 2 1 2 i j j
Now,for1≤α <α <α <α ≤12suchthatα +α (cid:54)=13,define
1 2 3 4 i j
Σ{α ,α ,α ,α }={(α ,α ,α ,α )∈I(4,12):α +α (cid:54)=13with1≤i,j ≤4}.
1 2 3 4 1 2 3 4 i j
Lemma1. WiththenotationabovewehaveapartitionofI(4,12),givenby
(cid:0) (cid:91) (cid:1) (cid:0) (cid:91) (cid:1)
C (Σ )∪ Σ{α ,α } ∪ Σ{α ,α ,α ,α } .
2 6 1 2 1 2 3 4
1≤α1<α2≤12 1≤α1<α2<α3<α4≤12
α1+α2(cid:54)=13 αi+αj(cid:54)=13
Proof. It is enough to show that every element in I(4,12) is included in one and only
one of the three different types of sets on the right hand side of the equality. Let α =
(α ,α ,α ,α ) ∈ I(4,12). For α +α (cid:54)= 13, where i,j = 1,2,3,4, it follows that
1 2 3 4 i j
(α ,α ,α ,α ) ∈ Σ{α ,α ,α ,α }. If for some α = (α ,α ,α ,α ) we have α +
1 2 3 4 1 2 3 4 1 2 3 4 1
α (cid:54)= 13, without loss of generality we may assume that α +α = 13, and then α ∈
2 3 4
Σ{α ,α }. Finally, if for some α = (α ,α ,α ,α ), α +α = α +α = 13, then
1 2 1 2 3 4 1 2 3 4
α∈C (Σ ). (cid:3)
2 6
Remark 1. For each 1 ≤ α < α ≤ 12 such that α + α (cid:54)= 13, we have that
1 2 1 2
|Σ{α ,α }| = 4 and |Σ{α ,α } : 1 ≤ α < α ≤ 12| = 60. Hence, in the second
1 2 1 2 1 2
term of the displayed expression in Lemma 1 there are 240 indexes. Also, for each 1 ≤
α <α <α <α ≤12suchthatα +α (cid:54)=13,wehavethat|Σ(α ,α ,α ,α )|=1,
1 2 3 4 i j 1 2 3 4
and|Σ{α ,α ,α ,α }:1≤α <α <α <α ≤12|=240. Hence,inthethirdterm
1 2 3 4 1 2 3 4
ofthedisplayedexpressioninLemma1thereare240indexes.
4 J.CARRILLO–PACHECO,F.JARQU´IN–ZA´RATE,M.VELASCO–FUENTES,ANDF.ZALDIVAR
We traslate now the combinatorial data of Lemma 1 in terms of the systems of linear
equationsassociatedtothecontractionmapf : ∧6E −→ ∧4E. Foreachα ∈ I(4,12)
rs
considerthelinearequation(2.2). Now,forthepartC (Σ )inLemma1,writing
2 6
C (Σ )={(P ,P ),(P ,P ),(P ,P ),(P ,P ),(P ,P ),(P ,P ),(P ,P ),
2 6 1 2 1 3 1 4 1 5 1 6 2 3 2 4
(P ,P ),(P ,P ),(P ,P ),(P ,P ),(P ,P ),(P ,P ),(P ,P ),(P ,P )}.
2 5 2 6 3 4 3 5 3 6 4 5 4 6 5 6
ForthesetC (Σ )orderedasabove, fillingthesymbols(cid:116)in(2.3)weobtainthesystem
2 6
oflinearequations
Π :=X +X +X +X =0
(P1P2) P3P1P2 P4P1P2 P5P1P2 P6P1P2
Π :=X +X +X +X =0
(P1P3) P2P1P3 P4P1P3 P5P1P3 P6P1P3
Π :=X +X +X +X =0
(P1P4) P2P1P4 P3P1P4 P5P1P4 P6P1P4
(3.1) Π :=X +X +X +X =0
(P1P5) P2P1P5 P3P1P5 P4P1P5 P6P1P5
Π :=X +X +X +X =0
(P1P6) P2P1P6 P3P1P6 P4P1P6 P5P1P6
Π :=X +X +X +X =0
(P2P3) P1P2P3 P4P1P3 P5P1P3 P6P1P3
. .
. .
. .
Π :=X +X +X +X =0.
(P5P6) P1P5P6 P2P5P6 P3P5P6 P4P5P6
whereweidentifythevariableX =X ifsupp{P P P }=supp{P(cid:48)P(cid:48)P(cid:48)}.
PiPjPk Pi(cid:48)Pj(cid:48)Pk(cid:48) i j k i j k
Similarly, for the second part in the partition of I(4,12) in Lemma 1, for the sets
Σ{α ,α }, for each 1 ≤ α < α ≤ 12, consider the system of four homogeneous
1 2 1 2
linearequationsΠ of(2.3),foreachα ∈Σ{α α },whichhavetheform:
αrs rs 1 2
X +X +X =0
1 2 3
(3.2) X +X +X =0
1 4 5
X +X +X =0
2 4 6
X +X +X =0.
3 5 6
and there are 60 such systems of linear equations (3.2). For example, for Σ{1,2} =
{12P ,12P ,12P ,12P },settingA = 10,B = 11andC = 12,asin(2.3),thesystem
3 4 5 6
(3.2)is
Π :=X +X +X =0
12P3 412P39 512P38 612P37
Π :=X +X +X =0
12P4 312P4A 512P48 612P47
Π :=X +X +X =0
12P5 312P5A 412P59 612P57
Π :=X +X +X =0.
12P6 312P6A 412P69 512P68
Finally,forthethirdpartinthepartitionofI(4,12)inLemma1,foreachsetΣ{α ,α ,α ,α },
1 2 3 4
with1 ≤ α < α < α < α ≤ 12,thematrixL ofthecorrespondinglinearequation
1 2 3 4 2
Π of (2.3) is a matrix (row vector) with the P and P
α1,α2,α3,α4 i,α1,α2,α3,α4 j,α1,α2,α3,α4
componentsequaltooneandalltheothercomponentsequaltozerofor1 ≤ i < j ≤ 6.
Thereare240suchmatricesL . Thesizeofthisvectoris1×C12. Forexample,
2 6
Π =X +X =0
1234 512348 612347
withcorrespondingmatrixL := (0,...,0,1,0,...,0,1,0,...,0),wherethefirst1isin
2
thecoordinatecorrespondingto123458andthesecond1isinthecoordinatecorreponding
tothe123467.
NOTEONTHECODIMENSIONOFTHELINEARSECTIONOFL(6,12) 5
Thecoefficientmatrixassociatedtothesystem(3.1)is
 
1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 
 
 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 
 
 0 0 1 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 
 
 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 
 
 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 
 
 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 
L4 = 0 0 1 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 
 
 0 0 0 1 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0 
 
 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 
 
 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 
 
 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1 1 0 
 
 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 
 
 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 1
andthecorrespondingmatrixassociatedtothesystem(3.2)is
 
1 1 1 0 0 0
L3 =10 01 00 11 10 01.
0 0 1 0 1 1
For the matrix L , adding its 14 last rows to the first row, we obtain that L is row-
4 4
(cid:18) (cid:19)
3
equivalenttothematrix L ,whereL(cid:98)4 isthematrixobtainedfromL4 deletingitsfirst
(cid:98)4
row,and3isarowallwhoseentriesareequalto3. Thus,ifchar(F) = 3,therankofL
4
is≤ 14. Moreover,adirectcomputationshowsthatitsrankisindeed14incharacteristic
3. Similarly, for the matrix L , adding the last three rows to the first one we see that if
3
char(F) = 2thentherankofL is≤ 3,andagainacomputationshowsthatisexactly3.
3
Moreover,rank(L )=4ifchar(F)(cid:54)=2,andrank(L )=15ifchar(F)(cid:54)=2,3.
3 4
Proposition2. ForanyfieldF,thematrixB,ofsizeC12×C12,associatedtothehomo-
4 6
geneoussystemΠ = {Π |α ∈ I(4,12)}canbegivenbyablockdiagonalmatrixas
αrs rs
follows
B =L ⊕(cid:16) (cid:77) L(α1,α2)(cid:17)⊕(cid:16) (cid:77) L(α1,α2,α3,α4)(cid:17),
4 3 2
1≤α1<α2≤12 1≤α1<α2<α3<α4≤12
α1+α2(cid:54)=13 αi+αj(cid:54)=13
 L 
4
 L3 0 
 
 ... 
B = 0 L3 ,
 L 
 2 
 
 ... 
L
2
wherethereare1matrixL ,60submatricesL ,and240submatricesL .
4 3 2
6 J.CARRILLO–PACHECO,F.JARQU´IN–ZA´RATE,M.VELASCO–FUENTES,ANDF.ZALDIVAR
Proof. ItfollowsfromtheobservationthatI(4,12)isadisjointunionofthesetsdescribed
in Lemma 1 and the one-to-one relationship between those sets and their corresponding
systemofhomogeneouslinearequations. (cid:3)
Forthecontractionmapf :∧6E −→∧4Egivenby(1.1),weobtain,fromProposition
2,thefollowingconsequences:
(1) Ifchar(F)=3,thenrank(B)=rank(L )+60rank(L )+240rank(L )=494.
4 3 2
(2) dimF(kerf)=C612−494=430.
(3) Ifchar(F)=3,thenthemapf isnotsurjective.
From Proposition 2 we calculate the codimension of the linear section P(kerf) in
P(∧6E)foranycharacteristic. Thatis
char(F) rank(B) codimensionofP(kerf)
0 495 429
2 430 494
3 494 430
p≥5 495 429
This computation shows that the codimension of P(kerf) in P(∧nE) depends on the
dimension2nofthesymplecticspaceE andthecharacteristicofthegroundfieldF. Ina
forthcomingpapertheauthorsshowthat,ingeneral,thecontractionmapf issurjectiveif
andonlyifchar(F)=0orchar(F)≥m,foracertainintegerm.
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DADAUTO´NOMADELACIUDADDEME´XICO,09390ME´XICO,D.F.,ME´XICO.
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E-mailaddress,J.Carrillo–Pacheco:[email protected]
E-mailaddress,F.Jarqu´ın–Za´rate:[email protected]
E-mailaddress,M.Velasco–Fuentes:[email protected]
E-mailaddress,F.Zaldivar:[email protected]