Table Of ContentThe degree of SO(n)
MadelineBrandt,DJBruce,TaylorBrysiewicz,RobertKrone,ElinaRobeva
7
1
0
2
n
a
J
AbstractWeprovideaclosedformulaforthedegreeofSO(n)overanalgebraically
3
closedfieldofcharacteristiczero.Inaddition,wedescribesymbolicandnumerical
1
techniqueswhichcanalsobeusedtocomputethedegreeofSO(n)forsmallvalues
] ofn.Asanapplicationofourresults,wegiveaformulaforthenumberofcritical
G
points of a low-rank semidefinite programming optimization problem. Finally, we
A providesomeevidenceforaconjectureregardingthereallocusofSO(n).
.
h
t
a
m 1 Introduction
[
2 ThespecialorthogonalgroupSO(n,R)isthegroupofautomorphismsofRnwhich
v preservethestandardinnerproductandhavedeterminantequaltoone.Thecomplex
0 specialorthogonalgroupisthecomplexificationofthespecialorthogonalgroupand
0
canbethoughtofmoreexplicitlyasthegroupofmatrices
2
3
0
MadelineBrandt
.
1 DepartmentofMathematics,UniversityofCalifornia,Berkeley,970EvansHall,Berkeley,CA,
0 94720e-mail:[email protected]
7
DJBruce
1
DepartmentofMathematics,UniversityofWisconsin,480LincolnDrive,Madison,WI,53706
:
v e-mail:[email protected]
i
X TaylorBrysiewicz
DepartmentofMathematics,TexasA&MUniversity,155IrelandSt,CollegeStation,TX77840
r e-mail:[email protected]
a
RobertKrone
DepartmentofMathematicsandStatistics,Queen’sUniversity,48UniversityAvenue,Kingston,
ON,K7L3N6e-mail:[email protected]
ElinaRobeva
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue,
Cambridge,MA02139e-mail:[email protected]
1
2 MadelineBrandt,DJBruce,TaylorBrysiewicz,RobertKrone,ElinaRobeva
SO(n):=SO(n,C)=(cid:8)M∈Mat (C)| detM=1, MtM=Id(cid:9).
n,n
Astheseconditionsarepolynomialsintheentriesofsuchamatrix,weviewSO(n)
asacomplexvariety.
RecallthatthedegreeofacomplexvarietyX isthegenericnumberofintersec-
tion points of X with a linear space of complementary dimension. Problem 4 on
Grassmannians in [Stu16] asks for a formula for the degree of the of SO(n). Our
primaryresultisthefollowingtheorem,whichanswersthisquestioncompletely.
Theorem1.1.ThedegreeofSO(n)isgivenby
(cid:18)(cid:18) (cid:19)(cid:19)
2n−2i−2j
degSO(n)=2n−1det .
n−2i
1≤i,j≤(cid:98)n(cid:99)
2
OurproofofTheorem1.1usesaformulaofKazarnovskij[Kaz87](seealsoThe-
orem2.4)forthedegreeoftheimageofarepresentationofaconnected,reductive,
algebraicgroupoveranalgebraicallyclosedfield.Byapplyingthisformulatothe
case of the standard representation of SO(n) we are able to express the degree of
SO(n)intermsofitsrootdataandotherinvariants.
Inadditiontothisresult,Theorem4.2providesacombinatorialinterpretationof
thisdegreeintermsofnon-intersectinglatticepaths.IncontrasttoTheorem1.1,the
combinatorialstatementhastheimmediatebenefitofbeingobviouslynon-negative.
Remark1.2.Let k be a field of characteristic zero. We can define SO(n,k) using
the same system of equations since they are defined over the prime field Q. For k
that is not algebraically closed, the degree of a variety can be defined in terms of
theHilbertseriesofitscoordinatering.SincetheHilbertseriesdoesnotdependon
thechoiceofk,thedegreedoesnoteither.WechoosetoworkoverCnotonlyfor
simplicity,butalsosothatwemayusetheabovedefinitionofdegree.
Remark1.3.Our methods are not specific to SO(n). The same approach can be
usedtocomputethedegreeofotheralgebraicgroups.Forexample,towardtheend
of Section 3 we provide a similar closed formula for the degree of the symplectic
group.ThisformulaisalsointerpretedcombinatoriallyinSection4.
InordertoverifyTheorem1.1,aswellasexplorethestructureofSO(n)infurther
depth,itisusefultocomputethisdegreeexplicitly.Wewereabletodothisforsmall
nusingsymbolicandnumericalcomputations.Acomparisonofthesuccessofthese
twoapproaches,togetherwithourformulafromTheorem1.1,isillustratedbythe
followingtable.
ThedegreeofSO(n) 3
n Symbolic Numerical Formula
2 2 2 2
3 8 8 8
4 40 40 40
5 384 384 384
6 - 4768 4768
7 - 111616 111616
8 - - 3433600
9 - - 196968448
Table1 DegreeofSO(n)computedinvariousways
Thisprojectstartedinthespringof2014,whenBenjaminRechtaskedthefifth
author to describe the geometry of low-rank semidefinite programming (see Sec-
tion5).Inparticular,heaskedwhytheaugmentedLagrangianalgorithmforsolving
this problem [BM05] almost always recovers the correct optimum despite the ex-
istenceofmultiplelocalminima.Itquicklybecameclearthattoevencomputethe
numberoflocalextrema,oneneedstoknowthedegreeoftheorthogonalgroup.In
Section5wefindaformulaforthenumberofcriticalpointsoflow-ranksemidefi-
niteprogramming(seeTheorem5.2).
The rest of this article is organized as follows. In Section 2 we give the reader
abriefintroductiontoalgebraicgroupsandstateKazarnovskij’sTheorem.Section
3provesTheorem1.1byapplyingKazarnovskij’sTheoremandsimplifyingthere-
sultingexpressions.Aftersimplification,weareleftwithadeterminantofbinomial
coefficients which can be interpreted combinatorially using the celebrated Gessel-
ViennotlemmawhichwedescribeinSection4.Therelationshipbetweenthedegree
ofSO(n)andthedegreeoflow-ranksemidefiniteprogrammingiselaboratedupon
in Section 5. Section 6 contains descriptions of the symbolic and numerical tech-
niques involved in the explicit computation of degSO(n). Finally, in Section 7 we
explorequestionsinvolvingtherealpointsonSO(n).
2 Background
Inthissectionweprovidethereaderwiththenecessarylanguagetounderstandthe
statement of Kazarnovskij’s Theorem (see Theorem 2.4), our main tool for deter-
miningthedegreeofSO(n).WeinvitethosewhoalreadyarefamiliarwithLiethe-
orytoskiptothestatementofTheorem2.4andcontinuetoSection3forourmain
result.Wenote,thatasidefromapplyingTheorem2.4,nounderstandingofthema-
terial in this section is necessary for understanding the remainder of the proof of
Theorem1.1.Amorethoroughtreatmentofthetheoryofalgebraicgroupscanbe
foundin[DK02,FH,Hum92].
An algebraic group G is a variety equipped with a group structure such that
multiplicationandinversionarebothregularmapsonG.Whentheunipotentradical
ofGistrivialandGisoveranalgebraicallyclosedfield,wesaythatGisareductive
4 MadelineBrandt,DJBruce,TaylorBrysiewicz,RobertKrone,ElinaRobeva
group. Throughout this section, we let G denote a connected reductive algebraic
groupoveranalgebraicallyclosedfieldk.LetG denotethemultiplicativegroup
m
ofk,soasaset,G =k\{0}.LetT denoteafixedmaximaltorusofG.Bymaximal
m
torus, we mean a subgroup of G isomorphic to Gr and which is maximal with
m
respecttoinclusion.Thenumberriswell-definedandiscalledtherankofG.After
fixing T, we define the Weyl group of G, denotedW(G), to be the quotient of the
normalizer of T by its centralizer,W(G)=N (T)/Z (T). Like r,W(G) does not
G G
dependonthechoiceofT uptoisomorphism.
Example2.1.WecanparametrizeSO(2,C)byG viathemap
m
1(cid:18) t+t−1 −i(t−t−1)(cid:19)
R(t):= ,
2 i(t−t−1) t+t−1
which is in fact a group isomorphism. (Note that R(eiθ) is the rotation matrix by
angleθ.)ThereforeSO(2)hasrank1.
Fixr∈N.Then
R(t ) 0 0··· 0
T2r:= 00...1 R(0...t2)00... ··.··.··. R(0...t )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)ti∈Gm∼=SO(2)r⊂SO(2r)
r
R(t ) 0 0··· 0 0
T2r+1:= 0...1 R(...t2)0... ·.·.·. 0... 0... (cid:12)(cid:12)(cid:12)(cid:12)ti∈Gm∼=SO(2)r⊂SO(2r+1)
00 00 00······ R(0tr)01 (cid:12)
are maximal tori of rank r of their respective groups. Therefore, rank(SO(2r))=
rank(SO(2r+1))=randweseethattherankofSO(n)dependsfundamentallyon
theparityofn.
ThecharactergroupM(T)isthesetofalgebraicgrouphomomorphismsfromT
toG ,i.e.grouphomomorphismsdefinedbypolynomialmaps,
m
M(T):=Hom (T,G ).
AlgGrp m
SinceT isisomorphictoGr ,allsuchhomomorphismsmustbeoftheform
m
(t ,...,t )(cid:55)→ta1···tar
1 r 1 r
with a ,...,a integers. This character group is isomorphic to Zr and for this rea-
1 r
sonitisoftencalledthecharacterlattice.Dualtothisisthegroupof1-parameter
subgroups
N(T):=Hom (G ,T),
AlgGrp m
ThedegreeofSO(n) 5
which is also isomorphic to Zr. Indeed, each 1-parameter subgroup is of the form
t (cid:55)→(tb1,...,tbr) for integers b1,...,br. There exists a natural bilinear pairing be-
tweenN(T)andM(T),givenby
M(T)×N(T)→Hom (G ,G )∼=Z
AlgGrp m m
(cid:104)χ,σ(cid:105)(cid:55)→χ◦σ.
Nowifρ:G→ GL(V)isarepresentationofGweattachtoitspecialcharacters
calledweights.Aweightoftherepresentationρ isacharacterχ ∈M(T)suchthat
theset
(cid:92)
V := ker(ρ(s)−χ(s)Id )
χ V
s∈T
is non-trivial. This condition is equivalent to saying that all of the matrices in
{ρ(s)|s∈T}haveasimultaneouseigenvectorv∈V suchthattheassociatedeigen-
valueforρ(s)is χ(s).WewilluseC todenotetheconvexhulloftheweightsof
V
therepresentationρ.
Example2.2.An example that will be important for us later will be the standard
representation coming from the natural embedding ρ :SO(n)→GL(Cn). For any
t∈G ,thematrixR(t)∈SO(2)haseigenvectorse +ie ande −ie witheigen-
m 1 2 1 2
valuest andt−1 respectively.FromtheexplicitdescriptionofT inExample2.1we
seethattheeigenvectorsofρ(t ,...,t )areallvectorsoftheforme ±ie with
1 r 2j−1 2j
1≤ j≤r andtheeigenvaluesaret±1,...,t±1.Theseeigenvalues,viewedaschar-
1 r
acters,aretheweightsofρ.Additionallywhenn=2r+1,wehavethate isan
2r+1
eigenvectorwitheigenvalue1,correspondingtothetrivialcharacter.
Another representation of a matrix group G⊆End(V) is the adjoint represen-
tation,Ad:G→GL(End(V)),withAd(g)thelinearmapdefinedbyA(cid:55)→gAg−1.
The roots of G are the weights of the adjoint representation. Given a linear func-
tional(cid:96)onM(T),wedefinethepositiverootsofGwithrespectto(cid:96)tobetheroots
χ suchthat(cid:96)(χ)>0.WedenotethepositiverootsofGbyα ,...,α.Forthealge-
1 l
braicgroupsinthispaper,wecanchoose(cid:96)tobetheinnerproductwiththevector
(r,r−1,...,1)sothatarootoftheforme −e ispositiveifandonlyif j<k.To
j k
eachrootα,weassociateacorootαˇ,definedtobethelinearfunctionαˇ(x):= 2(cid:104)x,α(cid:105)
(cid:104)α,α(cid:105)
where(cid:104),(cid:105)mustbeW(G)-invariant.Throughoutthispaper,wefixthistobethestan-
dardinnerproduct.
Example2.3.We now compute the roots of SO(n), starting with n even. It can be
shown that the simultaneous eigenvectors of Ad(s) over all s∈T are matrices A
withthefollowingstructure.Thesematricesarezerooutsidea2×2blockBinrows
2j−1,2jandcolumns2k−1,2kforsome1≤ j,k≤r.Furthermore,B=v vT with
1 2
each v equal to one of the eigenvectors of R(t), e ±ie . Indeed, suppose s∈T
k 1 2
has blocks along the diagonal R(t ) witht ,...,t ∈G . Then Ad(s)(A) will also
j 1 r m
bezeroexceptinthesame2×2block,andthatblockwillbe
R(t )BR(t )T =t±1t±1B,
j k j k
6 MadelineBrandt,DJBruce,TaylorBrysiewicz,RobertKrone,ElinaRobeva
wherethesignsintheexponentsdependonthechoicesofv andv .Thustheroots
1 2
ofSO(2r)arethecharactersoftheformt±1t±1for1≤ j,k≤r.
j k
Inthecasethatnisodd,Ahasanextrarowandcolumn.ConsiderAwithsupport
onlyinthelastcolumn.Thenfors∈T,Ad(s)(A)=sAs−1 buts−1 actstriviallyon
theleft,whilesactsonthelastcolumnasanelementofGL(Cn)asinthestandard
representation.AsinExample2.2wegetweightst±1,...,t±1,1.Thesameweights
1 r
appearforAwithsupportinthelastrow.
AssociatedtoGisaLiealgebrag,whichcomesequippedwithaLiebracket[, ].
ACartansubalgebrahisanilpotentsubalgebraofgthatisself-normalizing,mean-
ingif[x,y]∈hforallx∈h,theny∈h.LetS(h∗)betheringofpolynomialfunctions
on h. The Weyl groupW(G) acts on h, and this extends to an action ofW(G) on
S(h∗).ThespaceS(h∗)W(G) ofpolynomialswhichareinvariantuptotheactionof
W(G)isgeneratedbyrhomogeneouspolynomialswhosedegrees,c +1,...,c +1,
1 r
areuniquelydetermined.Thevaluesc ,...,c arecalledCoxeterexponents.
1 r
WearenowpreparedtostateKazarnovskij’stheorem.
Theorem2.4(Kazarnovskij’s Theorem, Prop 4.7.18 [DK02]). Let G be a con-
nected reductive group of dimension m and rank r over an algebraically closed
field.Ifρ :G→ GL(V)isarepresentationwithfinitekernelthen,
m! (cid:90)
degρ(G)= (αˇ αˇ ···αˇ )2dv.
|W(G)|(c1!c2!···cr!)2|ker(ρ)| CV 1 2 l
whereW(G) is the Weyl group, c are Coxeter exponents,C is the convex hull of
i V
theweights,andαˇ arethecoroots.
i
Ifρ isthestandardrepresentationforanalgebraicgroupG,thenitfollowsthat
degρ(G)=degG. Therefore, in order to compute degSO(n), all we must do is
apply this theorem for the standard representation of SO(n). The relevant data for
thistheoremisgiveninTable2belowforSO(n)andSp(n).
Table2
Group DimensionRank PositiveRoots Weights |W(G)| CoxeterExponents
SO(2r) (cid:0)22r(cid:1) r {ei±ej}i<j {±ei} r!2r−1 1,3,...,2r−3,r−1
SO(2r+1) (cid:0)2r2+1(cid:1) r {ei±ej}i<j∪{ei} {±ei} r!2r 1,3,5,...,2r−1
Sp(r) (cid:0)22r(cid:1) r {ei±ej}i<j∪{2ei} {±ei} r!2r 1,3,5,...,2r−1
ThedegreeofSO(n) 7
3 MainResult:TheDegreeofSO(n)
Wenowproveourmainresult,Theorem1.1.Attheendofthissectionweusethe
samemethodtoobtainaformulaforthedegreeofthesymplecticgroup.
WebeginbydirectlyapplyingTheorem2.4toSO(2r)andSO(2r+1)toobtain
(cid:18) (cid:19)
2r
! (cid:32) (cid:33)
2 (cid:90)
degSO(2r)= ∏ (x2−x2)2 dv, (1)
r!2r−1(r−1)!2r∏−1(2k−1)!2 CV 1≤i<j≤r i j
k=1
(cid:18) (cid:19)
2r+1
! (cid:32) (cid:33)
2 (cid:90) r
degSO(2r+1)= ∏ (x2−x2)2∏(2x)2 dv. (2)
r i j i
r!2r∏(2k−1)!2 CV 1≤i<j≤r i=1
k=1
Thus, to compute the degree of SO(n) it suffices to find formulas for the integrals
above. We do this by first expanding the integrand into monomials, and then in-
tegrating the result. We use the well-known expression for the determinant of the
Vandermondematrix,
(cid:32) (cid:33)
r
∏ (y −y)= ∑ sgn(σ)∏yσ(i)−1 .
j i i
1≤i<j≤r σ∈Sr i=1
Substitutingy =x2andsquaringtheentireexpressionyields
i i
(cid:32) (cid:33)
r
∏ (x2−x2)2= ∑ sgn(στ)∏x2σ(i)+2τ(i)−4 . (3)
i j i
1≤i<j≤r σ,τ∈Sr i=1
Additionally,wepointoutthateveryvariableintheintegrandisbeingraisedtoan
evenpowerandC istheconvexhullofweights,{±e}.Becauseofthissymmetry,
V i
theintegralsoverC are2rtimesthesameintegralsover∆ ,thestandardr-simplex.
V r
Wehavenowreducedthecomputationofthisintegraltounderstandingtheintegral
of any monomial over the standard simplex. The following proposition provides a
formulaforthis.
Proposition3.1(Lemma4.23[Mil14]).Let∆ ⊂Rr bethestandardr-simplex.If
r
a=(a ,...,a )∈Zr then
1 r >0
(cid:90) (cid:90) 1 r
xadx= xa1xa2···xardx dx ···dx = ∏a!.
∆r ∆r 1 2 r 1 2 r (r+∑ai)!i=1 i
Wecannowgetexpressionsfortheintegralsin(1)and(2)directlybyapplying
(3)andProposition3.1.
8 MadelineBrandt,DJBruce,TaylorBrysiewicz,RobertKrone,ElinaRobeva
Proposition3.2.LetI (r)andI (r)denotetheintegralsin(1)and(2)respec-
even odd
tively.Then,
r!2r
I (r)= det((2i+2j−4)!) .
even (cid:0)2r(cid:1)! 1≤i,j≤r
2
r!23r
I (r)= det((2i+2j−2)!) .
odd (cid:0)2r+1(cid:1)! 1≤i,j≤r
2
Proof. Asmentionedabove,wecancomputeI byconsideringtheintegrandonly
odd
overthesimplex.This,alongwithequation(3)givesusthat
(cid:90) r
I (r)=2r ∏ (x2−x2)2∏(2x)2dv
odd i j i
∆r1≤i<j≤r i=1
(cid:32) (cid:33)
(cid:90) r r
=2r ∑ sgn(στ)∏x2σ(i)+2τ(i)−4 ∏(2x)2dv
i i
∆r σ,τ∈Sr i=1 i=1
(cid:90) r
=23r ∑ sgn(στ) ∏x2σ(i)+2τ(i)−2dv.
i
σ,τ∈Sr ∆ri=1
Astheintegrandishomogeneousofdegree4(cid:0)r(cid:1)+2r,applyingProposition3.1and
2
simplifyingyields
23r r
Iodd(r)= (cid:0)4(cid:0)r(cid:1)+3r(cid:1)! ∑ sgn(στ)∏(2σ(i)+2τ(i)−2)!,
2 σ,τ∈Sr i=1
whichafterreplacingiwithσ−1(i)givesus
r r
∏(2σ(i)+2τ(i)−2)!=∏(2i+2τσ−1(i)−2)!.
i=1 i=1
Letρ=τσ−1.Overallpairsσ,τ∈S ,thepermutationρ appearsaseachpermuta-
r
tioninS exactlyr!times,andsgn(στ)=sgn(ρ).Therefore,wehavethat
r
r!23r r
Iodd(r)= (cid:0)4(cid:0)r(cid:1)+3r(cid:1)! ∑ sgn(ρ)∏(2i+2ρ(i)−2)!
2 ρ∈Sr i=1
r!23r
= det((2i+2j−2)!) .
(cid:0)2r+1(cid:1)! 1≤i,j≤r
2
ThederivationofI followspreciselythesamesteps. (cid:116)(cid:117)
even
Theorem1.1nowfollowsdirectlyfromthesubsequentsimplification.
ThedegreeofSO(n) 9
22r
degSO(2r+1)= det((2i+2j−2)!)
(1!3!···(2r−1)!)2
22r (cid:18)(2i+2j−2)!(cid:19)
= det
(1!2!···(2r−1)!) (2i−1)!
(cid:18) (cid:19)
(2i+2j−2)!
=22rdet
(2i−1)!(2j−1)!
(cid:18)(cid:18) (cid:19)(cid:19)
2i+2j−2
=22rdet .
2i−1
1≤i,j≤r
Reversing the order of the rows and columns of the final matrix and reindexing
producestheformulagiveninTheorem1.1.Similarly,fortheevencase,wehave
2
degSO(2r)= det((2i+2j−4)!)
(1!3!···(2r−3)!(r−1)!)2
2·(2r−1)2
= det((2i+2j−4)!)
(1!3!···(2r−3)!2·4···(2r−2))2
(cid:18) (cid:19)
(2i+2j−4)!
=22r−1det
(2i−2)!(2j−2)!
(cid:18)(cid:18) (cid:19)(cid:19)
4r−2i−2j
=22r−1det .
2r−2i
1≤i,j≤r
ThisfinishestheproofofTheorem1.1.
Since the orthogonal group O(n) has two components that are isomorphic to
SO(n),weimmediatelygetaformulaforthedegreeofO(n).
Corollary3.3.ThedegreeofO(n)isgivenby
(cid:18)(cid:18) (cid:19)(cid:19)
2n−2i−2j
degO(n)=2ndet .
n−2i
1≤i,j≤(cid:98)n(cid:99)
2
Furthermore,asmentionedintheintroduction,thereisnoreason,apriori,thatthe
stepstakeninthissectionareparticulartoSO(n).Wenowapplythesemethodsto
findthedegreeofSp(r),thegroupof(complex)symplecticmatrices.
RecallthesymplecticgroupoverCisdefinedtobe
Sp(r):=Sp(r,C)={M∈Mat (C)|MTΩM=Ω},
2r,2r
where
(cid:18) (cid:19)
0 I
Ω = r .
−I 0
r
Theorem3.4.ThedegreeofSp(r)isgivenby
(cid:18)(cid:18) (cid:19)(cid:19)
2i+2j−2
degSp(r)=det .
2i−1
1≤i,j≤r
10 MadelineBrandt,DJBruce,TaylorBrysiewicz,RobertKrone,ElinaRobeva
For 1≤r≤5 the values of degSp(r) are 2,24,1744,769408,2063048448,....
Thiswasverifiedusingbothnumericalandsymbolictechniquesuptor=3.
Proof. This is an application of Kazarnovskij’s result which is completely analo-
goustothecomputationforthespecialorthogonalgroup.Theintegralisthesame
astheoneforSO(2r+1)uptofactorsof2,soitisevaluatedinthesameway,and
thentheexpressioncanbesimplified
(cid:32) (cid:33)
(r(2r+1))! (cid:90) r
deg(Sp(r))= ∏ (x −x )2(x +x )2∏x2 dv
r!2r(1!3!···(2r−1)!)2 CV 1≤i<j≤r i j i j i=1 i
1
= det((2i+2j−2)!)
(1!3!···(2r−1)!)2 1≤i,j≤r
(cid:18)(cid:18) (cid:19)(cid:19)
2i+2j−2
=det .
2i−1
1≤i,j≤r
(cid:116)(cid:117)
WeremarkthatourformulafordegSp(r)isparticularlyinterestingbecausethe
determinant in Theorem 3.4 is the same as the determinant in Theorem 1.1 when
n=2r+1.
Corollary3.5.
degSO(2r+1)=22rdegSp(r)
Proof. Sendingthe(i,j)entryofthematrixinTheorem3.4tothe(r−i+1,r−j+1)
entrydoesnotchangethedeterminantandgivesusthat
(cid:18)(cid:18) (cid:19)(cid:19)
4r+2−2i−2j
degSp(r)=det .
2r+1−2i
1≤i,j≤r
Whenn=2r+1,thisisthematrixappearinginTheorem1.1andallthatisdifferent
isthecoefficientinfront.Accountingforthiscoefficientfinishestheproof. (cid:116)(cid:117)
4 Non-IntersectingLatticePaths
TheformulasgivenintheprevioussectionforthedegreesofSO(n),O(n),andSp(r)
canbeinterpretedasacountofnon-intersectinglatticepathsviatheGessel-Viennot
Lemma[GV85].
Lemma4.1(Gessel-Viennot(WeakVersion)).LetA={a ,...,a },B={b ,...,b }
1 r 1 r
be collections of lattice points in Z2. Let M be the number of lattice paths from
i,j
a tob usingonlyunitstepsineithertheNorthorEastdirection.Iftheonlyway
i j
thatasystemoftheselatticepathsfromA→Bdonotcrosseachotherisbysend-
inga (cid:55)→b,thenthedeterminantofM equalsthenumberofsuchnon-intersecting
i i
latticepaths.
