On seaweed subalgebras and meander graphs in type C

0.2 MB·

by  Dmitri Panyushev

July29,2016
ON SEAWEED SUBALGEBRAS AND MEANDER GRAPHS IN TYPE C
DMITRII.PANYUSHEVANDOKSANAS.YAKIMOVA
6
1
ABSTRACT. In2000,DergachevandKirillovintroducedsubalgebrasof”seaweedtype”in
0
2 gln(or sln) and computed their index using certain graphs. In this article, those graphs
l are called type-A meander graphs. Then the subalgebras of seaweed type, or just ”sea-
u
J weeds”,havebeendefinedbyPanyushev(2001)forarbitrarysimpleLiealgebras.Namely,

9 ifp1,p2 ⊂ gareparabolicsubalgebrassuchthatp1+p2 = g,thenq = p1∩p2 isaseaweed
2
ing. Ifp1andp2are“adapted”toafixedtriangulardecompositionofg,thenqissaidtobe

] standard. Thenumberofstandardseaweedsisfinite. Ageneralalgebraicformulaforthe
T
indexofseaweedshasbeenproposedbyTauvelandYu(2004)andthenprovedbyJoseph
R
(2006).
.
h
Inthispaper,elaboratingonthe“graphical”approachofDergachevandKirillov,wein-
t
a troducethetype-Cmeandergraphs,i.e.,thegraphsassociatedwiththestandardseaweed
m
subalgebrasofsp2n,andgiveaformulafortheindexintermsofthesegraphs.Wealsonote
[
thattheverysamegraphscanbeusedincaseoftheoddorthogonalLiealgebras.

2
RecallthatqiscalledFrobenius,iftheindexofqequals0. Weprovideseveralapplica-
v
5 tionsofourformulatoFrobeniusseaweedsinsp2n. Inparticular,usinganaturalpartition
0 of the set Fn of standard Frobenius seaweeds, we prove that #Fn strictly increases for
3
0 the passage from n to n+1. The similar monotonicity question is open for the standard
0 Frobeniusseaweedsinsln,evenforthepassagefromnton+2.
.
1
0
6
1. INTRODUCTION
1
:
v Theindexofan(algebraic)Liealgebraq,indq,istheminimaldimension ofthe stabilisers
i
X
forthecoadjointrepresentationofq. Itcanberegardedasageneralisationofthenotionof
r
a rank. Thatis,indqequalstherankofq,ifqisreductive. In[1],theindexofthesubalgebras
of ”seaweed type” in gl (or sl ) has been computed using certain graphs. In this article,
n n
those graphs are called type-A meander graphs. Then the subalgebras of seaweed type, or
just seaweeds, have been defined and studied for an arbitrary simple Lie algebra g [7].
Namely, if p ,p ⊂ g are parabolic subalgebras such that p +p = g, then q = p ∩p is a
1 2 1 2 1 2
seaweed in g. If p and p are “adapted” to a fixed triangular decomposition of g, then q
1 2
is said to be standard, see Section 2 for details. A general algebraic formula for the index
ofseaweedshasbeenproposed in [9, Conj.4.7]and then proved in [5, Section8].
2010MathematicsSubjectClassification. 17B08,17B20.
ThefirstauthorgratefullyacknowledgesthehospitalityofMPIM(Bonn)duringthepreparationofthe
article. ThesecondauthorispartiallysupportedbytheDFGpriorityprogrammeSPP1388“Darstellungs-
theorie”andtheGraduiertenkollegGRK1523“Quanten-undGravitationsfelder”.
1
2 D.PANYUSHEVANDO.YAKIMOVA
In this paper, elaborating on the “graphical” approach of [1], we introduce the type-C
meandergraphs,i.e.,thegraphsassociatedwith thestandard seaweedsubalgebrasofsp ,
2n
andgive aformula for the indexinterms ofthesegraphs. Although theseaweedsin sp
2n
are our primary object in Sections 2–4 , we note that the very same graphs can be used in
case ofthe odd orthogonal Liealgebras, see Section 5.
Recall that q is called Frobenius, if indq = 0. Frobenius Lie algebras are very important
in mathematics, because of their connection with the Yang-Baxter equation. We provide
some applications ofour formula to Frobenius seaweedsin sp . LetF denote the setof
2n n
standardFrobeniusseaweedsofsp . ForanaturalpartitionF = n F (seeSection4
2n n k=1 n,k
for details), we construct the embeddings F ֒→ F for all n,k > 1. Since F
n,k n+1,k+1 F n+1,1
does not meet the image of the induced embedding F ֒→ F and #(F ) > 0,
n n+1 n+1,1
this implies that #(F ) < #(F ). The similar monotonicity question is open for the
n n+1
standard Frobenius seaweeds in sl , even for the passage from n to n+2. We also show
n
that F andF are related to certain Frobenius seaweedsin sl .
n,1 n,2 n
Theground field isalgebraically closed and ofcharacteristic zero.
2. GENERALITIES ON SEAWEED SUBALGEBRAS AND MEANDER GRAPHS
Let p and p be two parabolic subalgebras of a simple Lie algebra g. If p +p = g, then
1 2 1 2
p ∩ p is called a seaweed subalgebra or just seaweed in g (see [7]). The set of seaweeds
1 2
includes all parabolics (if p = g), all Levi subalgebras (if p and p are opposite), and
2 1 2
many interesting non-reductive subalgebras. We assume that g is equipped with a fixed
−
triangular decomposition, so thatthere are two opposite Borel subalgebrasbandb ,and
−
a Cartan subalgebra t = b ∩ b . Without loss of generality, we may also assume that
− −
p ⊃ b (i.e., p is standard) and p = p ⊃ b (i.e., p is opposite-standard). Then the
1 1 2 2 2
−
seaweedq = p ∩p issaidtobestandard,too. Eitheroftheseparabolicsisdeterminedby
1 2
asubsetofΠ,thesetofsimpleroots associated with(b,t). Therefore, astandard seaweed
isdetermined bytwo arbitrary subsets ofΠ, see [7,Sect.2]for details.
For classical Lie algebras sl and sp , we exploit the usual numbering of Π, which al-
n 2n
lowsustoidentifythestandardandopposite-standardparabolicsubalgebraswithcertain
compositions related to n. Itisalsomore convenient todeal with gl in placeof sl .
n n
I. g = gl . Weworkwiththeobvioustriangulardecomposition ofgl ,wherebconsists
n n
of the upper-triangular matrices. If p ⊃ b and the standard Levi subalgebra of p is
1 1
gl ⊕...⊕gl ,thenwesetp = p(a),wherea = (a ,a ,...,a ). Notethata +···+a = n
a1 as 1 1 2 s 1 s
and all a > 1. Likewise, if p− ⊃ b− is represented by a composition b = (b ,...,b )
i 2 1 t
with b = n, then the standard seaweed p ∩ p− ⊂ gl is denoted by qA(a|b). The
j 1 2 n
A
corresponding type-A meandergraph Γ = Γ (a|b) isdefinedby the following rules:
P
• Γ hasnconsecutive vertices on a horizontal line numbered from 1up to n.
ONSEAWEEDSUBALGEBRASANDMEANDERGRAPHSINTYPEC 3
• The parts of a determine the set of pairwise disjoint arcs (edges) that are drawn
above the horizontal line. Namely, part a determines [a /2] consecutively embedded
1 1
arcs above the nodes 1,...,a , where the widest arc joins vertices 1 and a , the following
1 1
joins2anda −1,etc. Ifa isodd, thenthemiddlevertex(a +1)/2acquiresnoarcatall.
1 1 1
Next, part a determines[a /2]embeddedarcs above the nodesa +1,...,a +a , etc.
2 2 1 1 2
• The arcs corresponding to b are drawn following the same rules, but below the
horizontal line.
It follows that the degree of each vertex in Γ is at most 2 and each connected compo-
nent of Γ is homeomorphic to either a circle or a segment. (An isolated vertex is also a
segment!) By[1], the indexof qA(a|b) can be computed via Γ = ΓA(a|b) asfollows:
A
(2·1) indq (a|b) = 2·(numberof cycles in Γ)+(numberof segmentsin Γ).
Remark 2.1. Formula (2·1) gives the index of a seaweed in gl , not in sl . However, if
n n
q ⊂ gl isaseaweed,thenq∩sl isaseaweedinsl andtherespectivemappingq 7→ q∩sl
n n n n
isabijection. Hereq = (q∩sl )⊕(1-dim centre of gl ),henceind(q∩sl ) = indq−1. Since
n n n
indqA(a|b) > 1andthe minimal value‘1’isachievedifand onlyifΓ isasole segment, we
alsoobtain a characterisation of the Frobenius seaweedsin sl .
n
Example 2.2. ΓA(5,2,2|2,4,3)= ✛ ✘ and the index of the corre-
☛ ✟
✄ (cid:0) ✄ (cid:0)
r r r r r r r r r
✂ ✁ ✂ ✁
sponding seaweedin gl9 (resp. sl9)equa✒ls3 (resp✑. 2✡). ✠
II. g = sp . Weuse the embeddingsp ⊂ gl such that
2n 2n 2n
A B
sp = | A,B,C ∈ gl , B = Bˆ,C = Cˆ ,
2n C −Aˆ n
( ! )
where A 7→ Aˆ is the transpose with respect to the antidiagonal. If b˜ ⊂ gl (resp. b˜−) is
2n
the set of upper- (resp. lower-) triangular matrices, then b = b˜ ∩sp and b− = b˜− ∩sp
2n 2n
are our fixed Borel subalgebras of g = sp . If p ⊃ b, then the standard Levi subalge-
2n 1
bra of p is gl ⊕... ⊕ gl ⊕ sp , where a + ··· + a + d = n, all a > 1, and d > 0.
a1 as 2d 1 s i
Since d is determined by n and the ‘gl’ parts, p can be represented by n and the com-
1
−
position a = (a ,...,a ). We write p (a) for it. Likewise, if p is represented by another
1 s n 2
composition b = (b ,...,b ) with b 6 n, then p ∩p− is denoted by qC(a|b). To a stan-
1 t j 1 2 n
dard parabolic p = p (a) ⊂ sp , one can associate the parabolic subalgebra ˜p ⊂ gl
1 n 2nP 1 2n
that is represented by the symmetric composition a˜ = (a ,...,a ,2d,a ,...,a ) of 2n. In
1 s s 1
the matrix form, the standard Levi subalgebra of p˜ has the consecutive diagonal blocks
1
gl ,...,gl ,gl ,gl ,...,gl and, for the above embedding sp ⊂ gl and compati-
a1 as 2d as a1 2n 2n
ble triangular decompositions, one has p = p˜ ∩ sp (and likewise for p− ⊂ sp and
1 1 2n 2 2n
p˜− ⊂ gl ), see [7, Sect.5] for details. If a˜ and ˜b are symmetric compositions of 2n, then
2 2n
4 D.PANYUSHEVANDO.YAKIMOVA
the seaweed qA(a˜|˜b) ⊂ gl is said to be symmetric, too. The above construction provides
2n
a bijection between the standard seaweedsin sp and the symmetric standard seaweeds
2n
in gl (or sl ).
2n 2n
Wedefine the type-Cmeander graph ΓC(a|b) for qC(a|b) to be the type-A meandergraph
n n
ofthe corresponding symmetric seaweed˜q = ˜p ∩˜p− ⊂ gl . Formally,
1 2 2n
C A ˜
Γ (a|b) = Γ (a˜|b).
n
Weindicate belownewfeatures ofthese graphs.
C
• Γ (a|b) has2nconsecutive vertices on a horizontal linenumbered from 1 up to2n.
n
• Part a determines [a /2] embedded arcs above the nodes 1,...,a . By symmetry,
1 1 1
thesamesetofarcsappearsabovethevertices2n−a +1,...,2n. Next,parta determines
1 2
[a /2]embeddedarcsabovethenodesa +1,...,a +a andalsothesymmetricsetofarcs
2 1 1 2
above the nodes2n−a −a +1,...2n−a , etc.
1 2 1
• If d = n− a > 0, then there are 2d unused vertices in the middle, and we draw
i
d embedded arcs above them. This corresponds to part 2d that occurs in the middle of a˜.
P
Thearcscorrespondingtobaredepictedbythesamerules,butbelowthehorizontalline.
• A type-C meander graph is symmetric with respect to the vertical line between the
n-th and(n+1)-th vertices, and thesymmetry w.r.tthisline isdenotedby σ. Wealso say
that this line is the σ-mirror. The arcs crossing the σ-mirror are said to be central. These
′
are exactly the arcs corresponding tod = n− a and d = n− b .
i j
Our main result is the following formula for the index in terms of the connected com-
P P
C
ponents ofΓ (a|b):
n
1
C
(2·2) indq (a|b) = (numberofcycles)+ (number ofsegments thatare not σ-stable).
n 2
To illustrate this formula, we recall that, for the parabolic subalgebra p with Levi part
gl ⊕...⊕gl ⊕sp , we have indp = a1 + ... + as + d, see [7, Theorem5.5]. Here
a1 as 2d 2 2
p− = sp and the composition b is empty. On the other hand, the graph ΓC(a | ∅) has
2 2n (cid:2) (cid:3) (cid:2) (cid:3) n
n central arcs below the horizontal line corresponding to b = ∅. Hence each part a
i
gives rise to ai cycles and, if a is odd, to one additional segment, which is σ-invariant.
2 i
The middle part corresponding to sp gives rise to d cycles. This clearly yields the same
(cid:2) (cid:3) 2d
answer,cf. Example2.3. HencewealreadyknowthatEq.(2·2)iscorrect,ifqisaparabolic
subalgebra, i.e., if a = ∅ or b = ∅. Note also that indp = 0 if and only if d = 0 and all
a = 1,i.e., ifp = b.
i
Example 2.3. Here a = (2,3) and n = 7 (hence d = 2), and the σ-mirror is represented
by the vertical dotted line. It is easily seen that the only segment here is σ-stable and the
total numberof circles is4. (The circles are depicted byblue arcs). Henceindp = 4.
∗ ∗
Remark2.4. 1)Forbothgl andsp ,onehasq (a|b) ≃ q (b|a). Henceonecanfreelychoose
n 2n
whatcomposition isgoing to appearfirst.
ONSEAWEEDSUBALGEBRASANDMEANDERGRAPHSINTYPEC 5
✗ ✔
✎ ☞ ✎ ☞
ΓC(2,3|∅): ✞ ☎ ✞ ☎ ✞ ☎
7 s s s s s s s s s s s s s s
✝ ✆
✖ ✕
✫ ✪
✫ ✪
✫ ✪
✫ ✪
✫ ✪
Fig. 1. The meandergraph for a parabolicsubalgebra ofsp
14
∗
2)Itis alsotrue thatq (a|b) isreductive (i.e.,a Levi subalgebra)ifand only ifa = b.
Convention. Ifqisaseaweedineithersp orgl ,andthecorrespondingcompositions
2n 2n
are not specified,then the respective meandergraph isdenoted by ΓC(q)or ΓA(q).
∗
Remark 2.5. Let q be a seaweed in sp or gl . Then there is a point γ ∈ q such that the
2n n
stabiliser q ⊂ q is a reductive subalgebra, see [8]. A Lie algebra possessing such a point
γ
inthedualspaceissaidtobe(strongly)quasi-reductive[2],seealso[6,Def.2.1]. Oneofthe
main results of [2] states that if a Lie algebra q = LieQ is strongly quasi-reductive, then
∗
there is a reductive stabiliser Q (with γ ∈ q ) such that any other reductive stabiliser Q
γ β
∗
(with β ∈ q ) is contained in Q up to conjugation. In [6] this subgroup Q is a called a
γ γ
maximalreductivestabiliser,MRSfor short. For a seaweedq = qA(a|b), an MRSof q can be
A
described interms ofΓ (a|b)[6, Theorem 5.3]. Asimilardescription ispossible in type C
C
ifwe use Γ (a|b). Itwill appearelsewhere.
n
3. SYMPLECTIC MEANDER GRAPHS AND THE INDEX OF SEAWEED SUBALGEBRAS
Inthissection, weprove formula (2·2)ontheindexoftheseaweedsubalgebrasoftype C.
Let us recall the inductive procedure for computing the index of seaweeds in a sym-
plectic Lie algebra introduced by the first author [7]. Suppose that a = (a ,...,a ) and
1 s
b = (b ,...,b ) are two compositions with a 6 n and b 6 n. Then we consider the
1 t i j
C
standard seaweedq (a|b) ⊂ sp .
n 2n P P
Inductive procedure:
C
1.If either a or b is empty, then q (a|b) is a parabolic subalgebra and the index is com-
n
puted using [7, Theorem5.5](cf. also Introduction).
2.Suppose that both a and b are non-empty. Without loss of generality, we can assume
that a 6 b . By[7, Theorem5.2],indqC(a|b) can inductively be computed asfollows:
1 1 n
(i) Ifa = b ,then qC(a|b) ≃ gl ⊕qC (a ,...,a |b ,...,b ), hence
1 1 n a1 n−a1 2 s 2 t
indqC(a|b) = a +indqC (a ,...,a |b ,...,b ).
n 1 n−a1 2 s 2 t
(ii) Ifa < b , then
1 1
indqC (a ,...,a |b −2a ,a ,b ,...,b ) if a 6 b /2;
indqC(a|b) = n−a1 2 s 1 1 1 2 t 1 1
n indqC (2a −b ,a ,...,a |a ,b ,...,b ) if a > b /2.
( n−b1+a1 1 1 2 s 1 2 t 1 1
6 D.PANYUSHEVANDO.YAKIMOVA
(iii) Step2terminates when one of the compositions becomes empty, i.e., one obtains
a parabolicsubalgebra in asmallersymplectic Lie algebra, where Step1applies.
Remark 3.1. Iterating transformations of the form 2(ii) yields a formula that does not re-
C
quire considering cases, see [7, Theorem5.3]. Namely, if a < b , then indq (a|b) =
1 1 n
indqC (a′|b′), where a′ = (a ,...,a ), b′ = (b′,b′′,b ,...,b ), and b′ and b′′ are de-
n−a1 2 s 1 1 2 t 1 1
p a p+1
fined as follows. Let p be the unique integer such that < 1 6 . Then
p+1 b p+2
1
b′ = (p + 1)b − (p + 2)a > 0 and b′ = (p + 1)a − pb > 0. (If b′ = 0, then it has to
1 1 1 2 1 1 1
be omitted.)
Theorem 3.2. Let q = qC(a|b) be a seaweed in sp and ΓC(q) = ΓC(a|b) the type-C meander
n 2n n
graphassociatedwith q. Then
1
C C C
indq (a|b) = #{cyclesof Γ (a|b)}+ #{segmentsof Γ (a|b) thatare not σ-stable}.
n n 2 n
Proof. Our argument exploits the above inductive procedure. Let us temporarily write
T (a|b) for the topological quantity in the right hand side of the formula. Let us prove
n
that for the pairs of seaweeds occurring in either 2(i) or 2(ii) of the inductive procedure,
the required topological quantity behavesaccordingly.
Ifa = b and gl is a direct summand of q, then the index of qC (a ,...,a |b ,...,b )
1 1 a1 n−a1 2 s 2 t
C C
decreases by a ; on the other hand, Γ (a ,...,a |b ,...,b ) is obtained from Γ (q) by
1 n−a1 2 s 2 t
deleting 2 a1 cycles (and two segments, which are not σ-invariant in case a is odd).
2 1
Thisisin perfect agreementwith the formula.
(cid:2) (cid:3)
If a < b , then one step of sp-reduction for q is equivalent to two steps of gl-reduction
1 1
forthe meandergraphofΓA(˜q),where ˜qisthe corresponding symmetric seaweedin gl .
2n
These two “symmetric” steps are applied one after another to the left and right hand
sides of ΓA(˜q) = ΓC(q). According to [6, Lemma5.4(i)], the gl-reduction does not change
′ ′
the topological structure of the graph. HenceTn(a|b) = Tn−a1(a|b).
Since we have already observed (in Section 2) that our formula holds for the parabolic
subalgebras, the result follows. (cid:3)
Example 3.3. For the seaweed q (3,3|4,5) in sp , the recursive formula of Remark 3.1
10 20
yieldsthe following chainof reductions:
q = qC (3,3|4,5) ❀ qC(3|1,5) ❀ qC(1,1|5) ❀ qC(1|3,1) ❀ qC(∅|1,1,1).
10 7 6 5 4
The last term represents the minimal parabolic subalgebra of sp corresponding to the
8
uniquelongsimpleroot. TherespectivegraphsaregatheredinFigure 2. Itisreadilyseen
that both ends of the graphs undergo the symmetric transformations on each step; also
all the segmentsare σ-stable andthe total numberofcycles equals1. Thus, indq = 1.
One can notice that each reduction step consists of contracting certain arcs starting
from some end vertices of a meander graph. Clearly, such a procedure does not change
ONSEAWEEDSUBALGEBRASANDMEANDERGRAPHSINTYPEC 7
✬ ✩
✬ ✩
C ✗ ✔
Γ (3,3|4,5): ✎ ☞✎ ☞ ✎ ☞✎ ☞
10 ✞ ☎
s s s s s s s s s s s s s s s s s s s s
✝ ✆ ✝ ✆ ✝ ✆
✍ ✌ ✍ ✌
✖ ✕ ✖ ✕
✧ ✦ ✧ ✦
✬ ✩
✬ ✩
C ✗ ✔
Γ (3|1,5): ✎ ☞ ✎ ☞
7 ✞ ☎
s s s s s s s s s s s s s s
✝ ✆
✍ ✌ ✍ ✌
✧ ✦ ✧ ✦
✬ ✩
✬ ✩
C ✗ ✔
Γ (1,1|5):
6 ✞ ☎
s s s s s s s s s s s s
✝ ✆
✍ ✌ ✍ ✌
✧ ✦ ✧ ✦
✬ ✩
✬ ✩
C ✗ ✔
Γ (1|3,1):
5 ✞ ☎
s s s s s s s s s s
✝ ✆
✍ ✌ ✍ ✌
✬ ✩
✬ ✩
ΓC(∅|1,1,1): ✗ ✔
4 ✞ ☎
s s s s s s s s
✝ ✆
Fig. 2. The reduction steps fora seaweedsubalgebra ofsp
20
thetopologicalstructureofthegraph,andthisisexactlyhowLemma5.4(i)in[6]hasbeen
proved.
Example 3.4. In Figure 3, one finds the graph of a seaweed in sp of index 1. The seg-
16
mentsthat are not σ-stable are depicted byred arcs.
C ✗ ✔ ✗ ✔
Γ (3,4|5,3): ✎ ☞ ✎ ☞
8 ✞ ☎ ✞ ☎ ✞ ☎
s s s s s s s s s s s s s s s s
✍ ✌ ✍ ✌✍ ✌ ✍ ✌
✧ ✦ ✧ ✦
Fig. 3. Aseaweed subalgebra of sp with index1
16
8 D.PANYUSHEVANDO.YAKIMOVA
4. APPLICATIONS OF SYMPLECTIC MEANDER GRAPHS
In this section, we present some applications of Theorem 3.2. We begin with a simple
property ofthe index.
C ′ C
Lemma 4.1. If a < n and b < n, then indq (a|b) = (n − n) + indq (a|b), where
i j n n′
′
n = max{ a , b }.
i P j P
P C P ′
Proof. Here Γ (a|b) contains n − n arcs crossing the σ-mirror on the both sides of the
n
′
horizontal line. They form n − n central circles, and removing these circles reduces the
indexbyn−n′ andyieldsthe graph ΓC(a|b). (cid:3)
n′
Recall that a Lie algebra q is Frobenius, if indq = 0. In the rest of the section, we apply
Theorem3.2tostudyingFrobeniusseaweeds. Clearly,ifqC(a|b)isFrobenius, thenΓC(a|b)
n n
has only σ-stable segments and no cycles. Another consequence of Theorem 3.2 is the
following necessary condition.
Lemma4.2. If qC(a|b) isFrobenius, theneither a < nand b = n orviceversa.
n i j
P P
Proof. If a < n and b < n, then the index is positive in view of Lemma 4.1. If
i j
C
a = b = n, then there are no arcs crossing the σ-mirror. Therefore Γ (a|b) consists
i Pj P n
of two disjoint σ-symmetric parts, and the topological quantity of Theorem 3.2 cannot be
P P
equalto0. (Moreprecisely,inthesecondcaseqC(a|b)isisomorphictotheseaweedqA(a|b)
n
in gl , andindq > 1 for all seaweedsq ⊂ gl ,see Remark2.1.) (cid:3)
n n
Graphically,Lemma4.2meansthat,foraFrobeniusseaweed,onemusthavesomecen-
tral arcs (=arcs crossing the σ-mirror) on one side of the horizontal line in the meander
graph, and then there has to be no central arcs on the other side. The number of cen-
tral arcs can vary from 1 to n (the last possibility represents the case in which one of the
F
parabolics is the Borel subalgebra). Let denote the set of standard Frobenius sea-
n,k
weeds whose meander graph contains k central arcs. Then F = n F is the set of
n k=1 n,k
all standard Frobenius seaweeds in sp . If qC(a|b) lies in F , then so is qC(b|a). As we
2n n n,k F n
areinterestedinessentiallydifferentmeandergraphs,wewillnotdistinguishgraphsand
algebras corresponding to (a|b) and (b|a). Set F = #(F /∼) and F = #(F / ∼),
n,k n,k n n
where ∼is the corresponding equivalence relation. Then
2, n = 3;
1, n = 2;
F F F
n,n = 1, n,n−1 = , and n,n−2 = 4, n = 4;.
2, n > 3. 

 5, n > 5.
It follows from Lemma 4.2 that if qCn(a|b) ∈ Fn and bj = n, then the integer k such

that qC(a|b) ∈ F is determined as k = n − a . In Figure 4, one finds the meander
n n,k i P
graphs ofFrobenius seaweedsin sp with k = 1 and2.
14 P
ONSEAWEEDSUBALGEBRASANDMEANDERGRAPHSINTYPEC 9
C ✗ ✔ ✗ ✔
Γ (2,4|4,3):
7 ✞ ☎ ✞ ☎ ✞ ☎ ✞ ☎ ✞ ☎
s s s s s s s s s s s s s s
✝ ✆ ✝ ✆
✍ ✌✍ ✌
✖ ✕ ✖ ✕
C ✗ ✔
Γ (3,2|2,5): ✎ ☞ ✎ ☞
7 ✞ ☎ ✞ ☎ ✞ ☎
s s s s s s s s s s s s s s
✝ ✆ ✝ ✆
✍ ✌ ✍ ✌
✧ ✦✧ ✦
Fig. 4. Frobenius seaweedsubalgebras ofsp
14
Lemma 4.3. If q ∈ F , then ΓC(q) has exactly k connected components (σ-stable segments)
n,k
correspondingtothe centralarcs. Furthermore,the total number of arcsinΓC(q) equals 2n−k.
Proof. 1) Let A be the i-th central arc and Γ the connected component of ΓC(q) that
i i
contains A . EachΓ is aσ-stable segment.
i i
• IfΓ = Γ fori 6= j,thencontinuationsofA andA meetsomewhereinthelefthand
i j i j
half of ΓC(q). By symmetry, the same happens in the right hand half, which produces a
cycle. Hence the connected components Γ ,...,Γ must bedifferent.
1 k
• Assumethatthere existsyetanotherconnected component Γ . Thenitbelongsto
k+1
onlyonehalfofΓC(q). Bysymmetry, thereisalsothe“same”componentΓ intheother
k+2
halfofΓC(q). Thiswould imply thatindq > 0.
2) Since the graph ΓC(q) has 2n vertices and is a disjoint union of k trees, the number
ofedges(arcs) must be 2n−k. (cid:3)
Lemma 4.4. For any k > 1, there is an injective map F → F . Moreover, F > F ,
n,k n+1,k+1 n+1 n
thatis,thetotalnumberofFrobeniusseaweedsstrictlyincreasesunderthepassagefromnton+1.
Proof. For any q ∈ F (k > 1), we can add two new vertices in the middle of ΓC(q)
n,k
and connect them by an arc (on the appropriate side!). This yields an injective mapping
F → F for anyk > 1and thereby an injection i : F ֒→ F .
n,k n+1,k+1 n n n+1
Since F does not intersect the image of i , the second assertion follows from the
n+1,1 n
fact thatF > 0 for anyn > 0,see examplebelow. (cid:3)
n+1,1
Example. WepointoutanexplicitelementqC(a|b) ∈ F . Forn = 2k,onetakesa = (2k)
n n,1
andb = (1,2k−1). Forn = 2k+1,onetakesa = (2k)andb = (1,2k). Forn = 4,themeander
graph is: .
✞ ☎ ✞ ☎ ✞ ☎ ✞ ☎
s s s s s s s s
Proposition 4.5. (i✝) F✆or a✝fixe✆dm✝∈✆N, the numbers Fn,n−m stabilise for n > 2m+1. In other
words,Fn,n−m = F2m+1,m+1 for all n > 2m+1.
F F
(ii) Furthermore, = +1.
2m+1,m+1 2m,m
Proof. (i) Letq = qCn(a|b) ∈ Fn,n−m. Then si=1ai = mand tj=1bj = n. Consider the n-th
vertex of the graph (one that isclosest to the σ-mirror). Weare interested in b , the sizeof
P P t
10 D.PANYUSHEVANDO.YAKIMOVA
the last part of b, i.e., the part that contains the n-th vertex. By the assumption, we have
n−m central arcs over the horizontal line. Therefore, if n > 2m+2 and b > 2, then the
t
smallest arc corresponding to b hits two vertices covered by central arcs above the line.
t
Andthisproducesacycleinthegraph! Thiscontradiction showsthattheonlypossibility
is b = 1. Then one can safely remove two central vertices from the graph and conclude
t
that Fn,n−m = Fn−1,n−1−m as long as n > 2m+2. (The last step is opposite to one that is
used in the proof of Lemma4.4.)
(ii) Again, for q = qC (a|b) ∈ F , we consider b , the last coordinate of b. If
2m+1 2m+1,m+1 t
b = 1,thenthecentralpairofverticesinΓC(q)canberemoved,whichyieldsaseaweedin
t
F . Next, it iseasily seen that ifb ∈ {2,3,...,2m}, then ΓC(q) contains a cycle. Hence
2m,m t
this is impossible. While for b = 2m + 1, one obtains a unique admissible possibility
t
a = (1,1,...,1). (cid:3)
m
Rema|rk.{zUsin}ga similaranalysis, one can show that F2m,m = F2m−1,m−1 +3, ifm > 3.
F
Remark4.6. Ourstabilisationresultfor n,n−mcanbecomparedwith[3],whereDufloand
Yu consider a partition of the set of standard Frobenius seaweeds in sl into classes and
n
studytheasymptoticbehaviourofthecardinalityoftheseclassesasntendstoinfinity. Let
p(a)bethenumberofnonzeropartsofthecompositionaandletF˜ bethenumberofthe
n,p
A
standardFrobeniusseaweedsq (a|b)∩sl suchthatp(a)+p(b) = p. By[3,Theorem1.1(b)],
n
if n is sufficiently large, then F˜n,n+1−t is a polynomial in n of degree [t/2], with positive
rational coefficients.
F
Itseemsthat isthemostinterestingpartofthesymplecticFrobeniusseaweeds. Re-
n,1
callfromSection2thattoanystandardseaweedq ⊂ sp onecanassociatea“symmetric”
2n
seaweed˜q ⊂ gl such that q = ˜q∩sp . In thiscontext, we alsoset˜q = ˜q∩sl .
2n 2n 0 2n
Proposition 4.7. (i) If q ∈ F ,thenind˜q = 1, hence˜q isaFrobenius seaweedinsl .
n,1 0 2n
(ii) Thereisan injectivemapF → F , whichis notonto if n > 2.
n,1 n+1,1
Proof. (i) Ifq ∈ F ,thenΓC(q)andtherebyΓA(˜q)consistsofasolesegment(Lemma4.3).
n,1
ByEq. (2·1), we have ind˜q = 1 andtherefore ind˜q = ind˜q−1 = 0.
0
(ii) If q = qC(a|b) ∈ F , then s a = n − 1 and t b = n. We associate to it
n n,1 i=1 i j=1 j
a seaweed ˆq ∈ F as follows. Set ˆq = qC (aˆ|b), where aˆ = (a ,...,a ,2). Note that
n+1,1 P n+1 P 1 s
C C
Γ (a|b) has one central arc above the horizontal line, while Γ (aˆ|b) has one central arc
n n+1
below. The following isa graphical illustration ofthe transform q 7→ ˆq:
... ... 7→ ... ...
✞ ☎ ✞ ☎ ✞ ☎
s s s s s s
✆F ✝ ✆ ✝ ✆F ✝
This provides a bijection between and the seaweeds in whose last part of the
n,1 n+1,1
composition thatsumsupton+1equals2. Ifn+1 > 3,thenthereareseaweedsinF
n+1,1
such that the above-mentioned lastpart isbigger than 2. Hence F < F . (cid:3)
n,1 n+1,1