Meander Graphs and Frobenius Seaweed Lie Algebras II

We provide a recursive classification of meander graphs, showing that each meander is identified by a unique sequence of fundamental graph theoretic moves. This sequence is called the meander’s signature and can be used to construct arbitrarily large sets of meanders, Frobenius or otherwise, of any size and configuration. In certain special cases, the signature is used to produce an explicit formula for the index of seaweed Lie subalgebra of sl(n) in terms of elementary functions.


Introduction
Dergachev and A. Kirilov [1] introduced meanders as planar graph representations of bi-parabolic, or seaweed, subalgebras of sl(n) and also provided a combinatorial method of computing the index of such seaweeds from the number and type of the connected components of their associated meander graphs. Of particular interest are those seaweed algebras whose associated meander graph consists of a single path. Such algebras have index zero. More generally, algebras with index zero are called Frobenius and have been extensively studied in the context of invariant theory [2][3][4][5] and more recently from the point of view of quantum group theory [3].
Extending the work of Elashvilli, Coll et al. [2,6] showed that a seaweed of type | | a b c n was Frobenius precisely when gcd (a + b, b + c)=1. However, the methods used there do not extend to seaweeds with more than four blocks. Also, the question of what the index actually is for the non-Frobenius four block case was left unaddressed. We take up these questions in this follow-up note, by first providing a recursive classification of meander graphs, showing that each meander is identified by a unique sequence of fundamental graph theoretic moves.
This sequence is called the meander's signature. The signature provides a fast algorithm for the computation of the index of Lie algebra associated with the meander, and can therefore be used to test any relatively prime conditions based on the type of the meander that might define Frobenius seaweed Lie algebra. In particular, we find that an easy induction on the moves of the signature gives that the index of a seaweed of type | | a b c n is gcd (a + b, b + c)−1. The signature also yields, with relative ease, new infinite families of Frobenius seaweed Lie algebras of any size and type.

Definitions
In this section, we detail the notions of the index of Lie algebra, and seaweed algebras and the meanders associated with them.  be the standard basis in k n . A subalgebra of sl(n) that preserves the vector spaces {V i = span (e 1 , . . . , e a1 +…+a i )} and {W j = span (e b1 +…+b j +1, ..., e n )} is called a seaweed algebra of type

Index
due to their suggestive shape when exhibited in matrix form (Given in left hand side of Figure 1).

Remark.
A basis free definition is available but not necessary for the present discussion. Also, the notion of seaweed algebra has been extended to reductive algebras by Panychev [7].

Meanders
(Given in right hand side of Figure 1).

Recursive Classification and Winding Down
In this section, we show that any meander can be contracted or "Wound Down" to the empty meander through a sequence of graph- theoretic moves; each of which is uniquely determined by the structure of the meander at the time of move application. We find that there are five such moves, only one of which affects the component structure of the graph and is therefore the only move capable of modifying the index of the graph, here defined to be twice the number of cycles plus the number of paths minus one. Dergachev and Kirillov showed that the index of the graph is precisely the index of the associated seaweed subalgebra. Since the sequence of moves which contracts a meander to the empty meander uniquely identifies the graph, we call this sequence the meander's signature. Although developed independently, we find that the signature is essentially a graph theoretic recasting of Panyushev's reduction algorithm, which [7] was used to develop inductive formulas for the index of seaweeds in gl(n). Here these inductive formulas are expressed in terms of elementary functions, which are laid plain by the explicit nature of the signature.

Lemma 4 (Winding Down). If M is a general meander of type
the we have the following cases: 1. Flip (F) If a 1 < b 1 , then simply exchange a i for b i to get 5. Pure contraction (P) If a 1 > 2b 1 , then Note that the Winding Down moves can be reversed to create a set of "Winding Up" moves, which can be used to build all meanders, Frobenius and otherwise. Remark. Although not our concern here, one of the major questions for both open and closed meanders is their enumeration. Using a different family of index preserving operators on meanders graphs, Dufflo and Yu have recently developed an approach which can, in certain cases, effect this enumeration via polynomials [8].

The signature
We call the cases in Lemma 4 moves since they move one meander to another. Notice that in each of these moves except the Flip, the number of vertices in the graph is reduced. Also note that in each move except for the Component Elimination move, the component structure of the meander is maintained. Given a meander, there exists a unique sequence of moves (elements of the set {F, C(c), B, R, P}) which reduce the given meander down to the empty meander. Such a list is called the signature of the meander. Since the only move that changes the component structure is C(c), we get the following corollary, which provides a recursive classification of Frobenius meanders.

Theorem 5 (Recursive classification). A meander is Frobenius if and only if the signature contains no C(c) move except the very last move which must be C(1).
Example: As an example of the signature of a Frobenius meander, considers the meander of type Theorem 5 provides a fast algorithm for computing the signature of a meander. Indeed, from the signature, one can read off the index of the meander by simply adding the parameters used in the Component Elimination moves and subtracting 1. That is, if {c 1 , c 2 , . . . , c m } are the parameters used in the Component Elimination moves, then the index

Index of a Meander with 4 Blocks
The determination of whether seaweed sub algebra is Frobenius usually relies on a combinatorial argument. This section recasts known formulas for computing the index in terms of elementary functions. Theorem 8 provides a new addition to this class of formulas and is a strengthening of a result of Coll et al. [6].
In order to prove the next result, we make repeated use of the following easy fact that follows from the Euclidean Algorithm.

Fact 7. For integers a and b, gcd(a, b)=gcd(a, a + b).
Now, in the style of Theorem 6, we have the following pleasing result.

Theorem 8. A meander of type
Certainly an F move has no effect on the index of the meander so we may assume any meander with four blocks is in one of these forms or its flip, whichever is more convenient at the time.

Case 2. A Block Elimination move B is performed.
This case has two subcases. The first is where ( ) c n c and a=2c. Then  , an identical argument holds. Note that Theorem 6 can be reproven very easily using exactly the same approach. We now have the following easy corollary to Theorem 8, which is the main result [6].

New infinite families of Frobenius meanders
While known infinite families of Frobenius meanders are few, new such families can, in theory, be developed by routine applications of inductive formulas developed by Panychev [7]. However, we find that, in practice, and in keeping with our approach here, recognizing if seaweed is Frobenius directly from its block type, via a relatively prime condition, is difficult. To illustrate, we note that while it is known that Frobenius meanders can have an arbitrarily large number of blocks [7,9,10], it is non-trivial to show that they can have arbitrarily many blocks of arbitrarily large size.

Theorem 10. If a is even and gcd(a, b) = 1, then the meander of type
is Frobenius.
Remark. The proof relies on a lengthy counting argument. We relegate the proof to Appendix A.
The simplified Winding Down process can be used in cooperation with Theorem 10 to obtain the following more general family of biparabolics: Corollary 11. If a is even and gcd(a, b) = 1, then the meander of type is Frobenius where c=b + ka for some integer k.
Proof. Let a, b and c be as given and let ℓ be the number of blocks of size a on the bottom in this meander. By Theorem 10, the meander where the α i and β j are integer coefficients. Substantial empirical evidence suggests that this is not so. Using the signature, exhaustive simulations have shown that there is no set of integer coefficients, all with absolute value at most ten, which can be used to build such a relatively prime condition. While this in not conclusive, it is compelling.
And since the addition of blocks seems to only complicate matters, we are led to conjecture that no single, or even finite set of, relatively prime conditions are sufficient to classify Frobenius meanders with at least five blocks.

Ongoing work
The notion of a signature consisting of a deterministic sequence of index preserving graph theoretic moves can be applied to other families of Lie algebras. A forthcoming note will introduce the notion of a "symplectic meander" from which a finite set of relatively prime conditions can be used to identify Frobenius seaweed subalgebras of sp(2n) in the three and four block case. The so(2n) case seems to be similarly tractable [11]. However, the five block case appears to be a barrier to this "relatively prime" approach in all cases.

Appendix -Proof of Theorem 11
In this proof, we will be working with meanders which have multiple top blocks and a single bottom block, so we make use of the following simplified notation: Write a meander of type 1 2 | |…| n a a a b as a 1 |a 2 |…|a n . We also require a preliminary

Lemma 12. For an even integer a and an odd integer b, a meander with k blocks a|a|…|a|b has index m if and only if the meander a|a|… |a|(b + 2a) with the same number of blocks also has index m.
Proof. The proof of this lemma consists of simply showing that any path and cycle structure is preserved as the meander is transformed between the two meanders M and M′ described below. Let A 1 , A 2 , . . . , For each integer i with 1 ≤ i ≤ a/2, we may replace the path In particular, Lemma 12 means that, when considering whether a general meander of type a| a | . . . | a | b is Frobenius, we may assume b < 2a. Now, call a segment between consecutive vertices in the drawing of a meander an end-segment if there is an edge of the meander connecting the two vertices on either end of this segment. Call a segment a top-endsegment if the segment is an end-segment and the corresponding edge is a top-edge. A segment gets mapped by the meander by following either the bottom edges or the top edges on either side of the segment. By Theorem 8, we may assume there are at least 3 blocks of size a in this meander. If we suppose a=2, then by Lemma 12, we know b ≤ 3. Evidently, this meander is Frobenius. Thus, we may assume a ≥ 4. If there was a cycle, then the Jordan Curve Theorem implies that there exists a segment between vertices that does not map (following edges of the meander) to the exterior face. Since we show that this is not the case, there must not be a cycle in the meander. This proof involves considering the segments between vertices in the meander graph and showing that each top-end-segment must be mapped by the meander to the exterior face. We then consider any segment and show that it either maps to a top-end-segment or to the exterior face.
Intuitively, the proof makes use of the following heuristic: We visualize the meander as an object (much like a shell) with openings in the top between the blocks. If water is poured into these openings, we ask whether the water permeates all areas inside the shell. If so, the meander is Frobenius and if not, it contains a cycle and is therefore not Frobenius.
Label the segments between consecutive vertices of the meander in the natural (drawn) order from 1 up to ka + b − 1 where k is the number of even blocks. Let c=a/2 and note that, since c ≥ 2, c also shares no common factor with b. Then any top-end-segment must be labeled with ωc where ω is an odd positive integer. Furthermore, the exterior face is accessible via any segment labeled ia = 2ic for any positive integer i ≤ k.
For any segment labeled x, following the bottom mapping by the meander yields 2kc+b−x since c = a/2 . We denote this by When following the top mapping by the meander, we have two cases. If we are in the odd group (of size b), we would map x to 4kc + b − x so we denote this by 4 .
Suppose now we map within an even group (of size a). If we start in the ℓ th such group, we would map x to (2ℓ − 1)2c − x which is denoted by These maps will be called arrow maps.
Our first goal is to show that, starting with a top-end-segment labeled ωc and following the mapping by the meander, we would reach the exterior face (represented by a segment labeled 2ic for some positive integer i as above) before ever reaching another top-endsegment. In order to accomplish this, we show that any sequence of these arrow maps would send ωc to 2ic before ever reaching ω′c for some positive odd integer ω′. After the first bottom map, we get where x is a positive integer.
In either case, the result is an odd multiple of c plus a multiple of b where the multiple of b never decreases and increases by at most one after double mapping. Since this double mapping can be repeated, we see that ωc will only map to ω′c + mb where ω′ is odd and m is a positive integer.