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^{1}Department of Mathematics, Lehigh University, Bethlehem, PA, USA

^{2}Department of Mathematics, Pace University, NY, USA

- *Corresponding Author:
- Vincent Coll

Department of Mathematics

Lehigh University, Bethlehem

PA, USA267-471-5320

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**E-mail:**[email protected]

**Received Date:** February 28, 2017; **Accepted Date:** April 18, 2017; **Published Date:** April 28, 2017

**Citation: **Coll V, Dougherty A, Hyatt M, Mayers N (2017) Meander Graphs and
Frobenius Seaweed Lie Algebras III. J Generalized Lie Theory Appl 11: 266. doi:
10.4172/1736-4337.1000266

**Copyright:** © 2017 Coll V, et al. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.

**Visit for more related articles at** Journal of Generalized Lie Theory and Applications

We investigate properties of a Type-A meander, here considered to be a certain planar graph associated to seaweed subalgebra of the special linear Lie algebra. Meanders are designed in such a way that the index of the seaweed may be computed by counting the number and type of connected components of the meander. Specifically, the simplicial homotopy types of Type-A meanders are determined in the cases where there exist linear greatest common divisor index formulas for the associate seaweed. For Type-A seaweeds, the homotopy type of the algebra, defined as the homotopy type of its associated meander, is recognized as a conjugation invariant which is more granular than the Lie algebra's index.

Lie algebra; Seaweed; Biparabolic; Meander; Rank; Index; Frobenius

The *index* of a Lie algebra is an important invariant of the Lie algebra and is bounded by the algebra's rank: ind , with equality when is reductive. More formally, the index of a Lie algebra is given by

where *f* is a linear functional on and *B _{f}* is the associated skew-symmetric

In [4] Dergachev and A. Kirillov introduced a combinatorial method for computing the index of certain seaweed (biparabolic) subalgebras of , based on counting the number and type of connected components of a planar graph representation of the seaweed algebra, called a (Type-A) *meander*. A Type-C, or *symplectic* meander and an attendant combinatorial index formula was developed by Coll et al in [2] for seaweed subalgebras of . (See also [5]).

Using a collection of deterministic graph theoretic moves, a given meander can be “wound down" to reveal its (simplicial) *homotopy type*. The sequence of moves used in this winding-down procedure is called the *signature* of the meander and may be regarding as a graph theoretic rendering of Panyushev's well-known reduction [6]. In [7,8], Coll et al used the signature to develop closed form formulas for the index of a seaweed algebra in terms of the block sizes of the defining flags in the Type-A and Type-C cases when the number of blocks in the flags is small. Subsequently, Karnauhova and Liebsher [10] used signature type moves and complexity arguments to establish that the index formulas developed in these papers are the only linear greatest common divisor formulas for the index based on the flags defining the seaweed. One finds that in the Type-A case, a seaweed is Frobenius precisely when its associated meander consists of a single path [4,7,8]. For a Type-C seaweed to be Frobenius, its associated meander must reduce to a certain collection of paths [5,10].

Since the homotopy type is not defined in terms of the algebraic structure of the original Lie algebra, it is not *a priori* clear that it is an algebraic invariant of the original algebra. In fact, this remains an open question for an arbitrary seaweed. However, it is implicit in the work of Moreau and Yakimova [11] that for Type-A seaweeds the homotopy type is a conjugation invariant which is more granular than the index. The example at the end of this paper provides two seaweed algebras which have the same dimension, rank, and index − but different homotopy types. So, are not conjugate.

In Theorems 5.1 and 5.2, we classify those homotopy types of seaweeds where there exist linear greatest common divisor index formulas developed in the prequels to this article.

Let q and q′ be two parabolic subalgebras of a reductive Lie algebra g. If q + q′ = g then q∩q′ is called a *seaweed*, or in the terminology of A. Joseph [13] *biparabolic*, subalgebra of . In what follows, we further assume that is simple and comes equipped with a triangular decomposition

where is a Cartan subalgebra of and and are the subalgebras consisting of the upper and lower triangular matrices, respectively. Let Π be the set of 's simple roots and for *β*∈Π, let denote the root space corresponding to *β*. A seaweed subalgebra q∩q′ is called standard if and We tacitly assume that the ground field is an algebraically closed field of characteristic zero, so that any seaweed is conjugate to a standard one. Note that while two standard parabolic subalgebras cannot be conjugate, two standard seaweeds can be.

In the case that q∩q is standard, let , , and denote the seaweed by . Such a seaweed is parabolic if one of Ψ or Ψ′ or is the empty set, and called maximal parabolic if it is of the form , respectively.

Let be the algebra of *n*×*n* matrices with trace zero and consider the triangular decomposition of as above. Let = {*β*_{1},…, *β _{n}*

Now, following the notational conventions established in [3], define the type of the seaweed to be the symbol

By construction, the sequence of numbers in __a__ determines the heights of triangles below the main diagonal in , which may have nonzero entries, and the sequence of numbers in __ b__ determines the heights of triangles above the main diagonal. For example, the seaweed of type has the following shape, where * indicates the possible nonzero entries from the ground field. See

**Type-A meanders**

Following Dergechev and A. Kirillov [5], we associate a planar graph to each seaweed of type as follows. Line up *n* vertices horizontally and label them . Partition the set of vertices into two set partitions, called top and bottom. The top partition groups together the first a_{1} vertices, then the next a_{2} vertices, and so on, lastly grouping together the last *a _{m}* vertices. In a similar way, the bottom partition is determined by the sequence

Evidently, every meander consists of a disjoint union of cycles, paths, and points (degenerate paths). The main result of [5] is that the index of the meander can be computed by counting the number and type of each of these components.

**Theorem 3.1** (**Theorem 5.1,** [5]): If p is a seaweed subalgebra of , then

where *C* is the number of cycles and *P* is the number of paths in the associated meander.

This elegant result, and the Type-C analogue ([Theorem 4.5) are difficult to apply in practice. However, in certain cases, the following index formulas allow us to ascertain the index directly from the block sizes of the flags that define the seaweed. The following formulas were developed in the first two articles in this series. We hasten to add that the formula in Theorem 3.2, which follows as a corollary to Theorem 3.3, was known early on to Elashvilli [14].

**Theorem 3.2** (**Theorem 7**, [3]): A seaweed of type has index gcd(*a*; *b*)−1

**Theorem 3.3** (**Theorem 8**, [3]): A seaweed of type , or type , has index gcd(*a* + *b*; *b* + *c*) − 1.

The following result establishes that the formulas in Theorems 3.2 and 3.3 the only nontrivial linear ones that are available in the parabolic case.

**Theorem 3.4** (**Theorem 5.3**, [11]): If *m* ≥ 4 and p is a seaweed of type, then there do not exist homogeneous polynomials *f*_{1}, of arbitrary degree, such that the index of is given by gcd(*f*_{1}(*a*_{1},…, *a _{m}*),

**Homotopy type**

**Definition 3.5:** We say that a planar graph has *homotopy type* *H*(*a*_{1}, *a*_{2},…,*a _{m}*) if its homotopy type is equivalent to the meander of type . That is, a union of

**Example:** A planar graph with homotopy type *H*(5, 1, 2) is homotopically equivalent to the graph in the following **Figure 3**.

When it makes sense, we define the *homotopy type of* a *seaweed* to be the homotopy type of its corresponding meander. More pointedly, we note that, unlike the index, the homotopy type of a Lie algebra g is not defined directly in terms of g's Lie Theory. It is therefore not *a priori* clear to what extent the homotopy type is an algebraic invariant. However, the following theorem follows from Theorem 5.3 in the recent paper by Moreau and Yakimova [11].

**Theorem 3.6:** Conjugate seaweed subalgebras of sl(n) have the same homotopy type.

In this section, we recall two technical Lemmas See [8,9]. The first Lemma (Winding Down) can be used to discern the homotopy type of a meander. The second Lemma (Winding Up), in cooperation with the Winding-Down Lemma, will be used in the proof of the classification Theorems 5.1 and 5.2. In these theorems, we classify the homotopy types in the cases where there exist linear greatest common divisor index formulas for the associated seaweed.

The Winding-Down Lemma establishes that, through a deterministic sequence of graph theoretic moves, each meander can be contracted or “wound down” to reveal its homotopy type. Since the sequence of moves is uniquely determined by the meander, we refer to this sequence as the meander's signature. Essentially, the winding-down moves, and the attendant signature, are a graph-theoretic recasting of Panyushev's reduction algorithm which was used in [14] to develop inductive formulas for the index of seaweeds in .

**Lemma 4.1 (Winding Down):** Given a meander *M* of type , create a meander *M*′ by one of the following moves. For all moves except the Component Elimination, *M* and *M*′ have the same homotopy type.

1. Flip (*F*): If *a*_{1} < *b*_{1}, then

2. Component Elimination (*C*(*c*)): If *a*_{1} = *b*_{1} = *c*, then of type.

3. Rotation Contraction (*R*): If *b*_{1} < *a*_{1} < 2*b*_{1}, then of type.

4. Block Elimination (B): If *a*_{1} = 2*b*_{1}, then of type .

5. Pure Contraction (P): If *a*_{1} > 2*b*_{1}, then of type.

**Example:** We continue with our running example, and wind down the meander of **Figure 2** in **Figure 4** below.

Note that each of the winding-down moves can be reversed to yield a winding-up move. The winding-up moves, which we record in the following Lemma, can be used to build up any meander, of any size and block configuration.

**Lemma 4.2 (Winding Up):** Every meander is the result of a sequence of the following moves applied to the empty meander. For all moves, except Component Creation, *M* and *M*′ have the same homotopy type.

Given a meander *M* of type , create a meander *M*′ by one of the following moves:

1. Flip of type .

2. Component Creation : If *a*_{1} = *b*_{1} = *c*, then of type.

3. Rotation Expansion : If *b*_{1} <*a*_{1}< 2*b*_{1}, then of type.

4. Block Creation of type .

5. Pure Expansion of type.

In this section, we present the promised homotopy type classification theorems. The proofs are inductive in nature and based on the Winding-Up lemma.

**Theorem 5.1.** If is a seaweed subalgebra of of type or , with gcd(*a*,*b*) = *k*, then its homotopy type is *H*(*k*).

Proof. Given any integer *m*≥2, we prove by induction that the theorem holds for all such seaweeds created using m winding-up moves. For the base of induction, the only seaweeds of this type that can be created by two moves are those created by Component Creation followed by Block Creation. Such a meander has type . The homotopy type is *H*(*a*), and gcd(*a*,*a*) = *a*.

Now, let *m* > 2 and assume the theorem holds for all such seaweeds created using *m*−1 winding-up moves. Let p be a seaweed of type or created by *m*-1 winding-up moves. Also let *k* = gcd(*a*, *b*). Let be the seaweed obtained from p by applying the appropriate windingdown move, so that is created using *m*-1 winding-up moves. We break the proof into cases depending on which winding-down move is applied. For all cases except Component Elimination, and have the same homotopy type.

**Case 1:** A Flip is applied to .

This case is trivial.

**Case 2:** A Component Elimination is applied to .

This case is vacuously true − a Component Elimination cannot be applied to a seaweed of type or .

**Case 3:** A Rotation Contraction is applied to .

In order to apply this move, must have type with *n* < 2*a*, thus has type . For any positive integers *x* and *y* we have gcd(*x*, *y*) = gcd(*x*, *x* + *y*). Therefore,

gcd(*b*, 2*a*−*n*) = gcd(*b*, *b* + 2*a* −*n*) = gcd(*b*, *a*) = *k*.

The claim then follows from the inductive hypothesis.

**Case 4:** A Block Elimination is applied to .

In order to apply this move, must have type with *n*=2*a*, thus *b* = *a* and *k* = gcd(*a*; *b*) = *a*. The claim now follows from the fact that has type .

**Case 5:** A Pure Contraction is applied to

In order to apply this move, must have type with *n*>2*a*, thus has type. Now

gcd(*a*, *n*−2*a*) = gcd(*a*, *n*−2*a* + *a*) = gcd(*a*, *b*) = *k*.

So, the claim follows from the inductive hypothesis.

**Theorem 5.2.** Let seaweed subalgebra of of type, . Let *k* = gcd(*a* + *b*, *b* + *c*), and let *r* ≡*a* mod *k*, and *s*≡*b* mod *k*, where 0 ≤*r* < *k* and 0≤*s* < *k*.

1. If *r* = 0 or *s* = 0, then the homotopy type is *H*(*k*).

2. If *r* and *s* are nonzero, then the homotopy type is *H*(*r*, *s*).

Proof. Note that k divides (*a* + *b*)−(*a* + *c*) = *a*−*c*, so *r* ≡ *a* ≡ *c* mod *k*. In the case that a seaweed has type , we have 2*n* = (*b* + *c*) + (*a* + *d*) and so, *k* = gcd(*n*, *b* + *c*) = gcd(*n*, *a* + *d*). Also, *k* divides (*a* + *b*)−(*a* + *d*) = *b*−*d*, so *s* ≡ *b* ≡ *d* mod *k*.

Given any integer *m*≥2, we prove, by induction, that the theorem holds for all such seaweeds created using *m* winding-up moves. For the base of induction, the only seaweeds of this type that can be created by two moves are those created by two Component Creation moves. Such a meander has type and the homotopy type is *H*(*a*, *b*). Since *k* = gcd(*a* + *b*, *b* + *a*) = *a* + *b*, we have *r* = *a* and *s* = *b*, as desired.

Now, let *m* > 2 and assume the theorem holds for all such seaweeds created using *m*−1 winding-up moves. Let be a seaweed of type , created by m winding-up moves. Also, let *k* = gcd(*a* + *b*, *b* + *c*) and be the seaweed obtained from p by applying the appropriate winding-down move, so that is created using *m*−1 winding-up moves. We break the proof into cases depending on which windingdown move is applied. For all cases except Component Elimination, and have the same homotopy type.

**Case 1:** A Flip is applied to .

This case is trivial.

**Case 2:** A Component Elimination is applied to .

To apply this move, must have type . As noted above, the homotopy type is *H*(*r*, *s*), as desired.

**Case 3:** A Rotation Contraction is applied to .

If has type , then this move cannot be applied.

If has type , then we must have *n* < 2*a* in order to apply this move, and has type . Note that

gcd(*c*+*b,* *b*+2*a*−*n*) = gcd(*b*+*c*, *a*−*c*) = gcd(*b*+*c*, *b*+*c*+*a*−*c*) = *k*.

So, by induction, has homotopy type *H*(*k*) or *H*(*r*, *s*) depending on the values of *r* ≡ *c* mod *k* and *s* ≡ *b* mod *k*, respectively.

If has type , then we must have *c* < *a* < 2*c* in order to apply this move, and has type.

Now, gcd(*c*+*b*, *c*+*d*) = gcd(*c*+*b*, *n*) = *k*, so, by induction, has homotopy type *H*(*k*) or *H*(*r*; *s*) depending on the values of *r* ≡ *c* mod *k* and *s* ≡ *d* mod *k*, respectively.

**Case 4:** A Block Elimination is applied to .

If has type, then this move cannot be applied.

If has type, then we must have *n* = 2*a* in order to apply this move. Therefore, *b* + *c* = *a* = *n*/2. Since has type, we know by Theorem 5.1 that the homotopy type is *H*(*j*) where *j* = gcd(*c*,*b*), but since *j* divides *b*+*c* = *a*, we also see that *j* divides *a*+*b*, so *j*≤*k.* Now, since *k* = gcd(*a*+*b*, *b*+*c*) = gcd(*n*/2+*b*, *n*/2), we have that *k* divides *b*, and so *k* also divides *a* and *c*. This implies *k* ≤ *j*. We conclude that *k* = *j*. Note that *r* = *s* = 0 and the homotopy type of is *H*(*k*), as desired.

If has type , then we must have *a* = 2*c* in order to apply this move, and has type. Again, by Theorem 5.1, the homotopy type is *H*(*j*) where *j* = gcd(*c*, *b*). Clearly, *j* divides *b* + *c*, and *j* also divides *a* + *b* = 2*c* + *b*, so *j*≤*k*. Now, *k* divides (*a* + *b*)−(*b* + *c*) = *a*−*c* = *c*, so *k* also divides *b* and *a*. This implies that *k*≤*j*. We conclude that *k* = *j*. Once again *r* = *s* = 0 and the homotopy type of is *H*(*k*), as desired.

**Case 5:** A Pure Contraction is applied to

If has type , then this move cannot be applied.

If has type , then we must have *n* > 2*a* in order to apply this move, and has type. Now gcd(*b* + *c*, *a* + *b*) = gcd(*b* + *c*, *n*) = *k*, so, by induction, has homotopy type *H*(*k*) or *H*(*r*, *s*) depending on the values of *r*≡*c* mod *k* and *s* ≡*b* mod *k*, respectively.

If has type , then we must have *a* > 2*c* in order to apply this move, and has type. Note that

gcd(*a*−2*c+c*, *c*+*b*) = gcd(*a*−*c*, *b*+*c*) = gcd(*a*−*c*+*b*+*c*, *b*+*c*) = *k*.

So, by induction, has homotopy type *H*(*k*) or *H*(*r*, *s*) depending on the values of *r* ≡ *c* mod *k* and *s* ≡*b* mod *k*, respectively.

The following example demonstrates that the homotopy type can sometimes distinguish between algebras when grosser invariants are unable to detect differences.

**Example:** The seaweeds of type and both have dimension (27), rank (7), and index (1), but have homotopy types *H*(1, 1) and *H*(2), respectively.

We define a meander to be a planar graph associated to a seaweed algebra which is designed in such a way that the index of the seaweed may be computed by counting the number and type of connected components of the graph. Such combinatorial formulas have been established in the Type-A case by Dergachev and A. Kirillov [4] and in the Type-C case by Coll et al. [11] (and independently by Panyushev and Yakimova [5]). These elegant formulas are difficult to apply in practice, but in certain cases can be replaced by closed form linear greatest common divisor formulas based on the flags that define the seaweed in its standard representation. For maximal parabolics in the Type-A case, this was done early on by Elashvili, and later, as a corollary to their combinatorial result, by Dergachev and A. Kirillov, and still later by Coll et al., as a consequence of a more generalized formula. The Type-C case was addressed by Coll et al. in [10]. Investigations by Karnauhova and Liebsher [13] show that these formulas are the full complement of linear greatest common divisor index formulas available in the Type-A and Type-C cases.

The simplicial homotopy type of a meander is evidently welldefined. Recent work of Moreau and Yakimova [12] establishes that the homotopy type of a Type-A seaweed is also well-defined, up to conjugation. In a forthcoming note, and after the fashion of this article, we will consider the Type-C case. In particular, we will establish that the homotopy type of a Type-C seaweed is a conjugation invariant. Furthermore, we will classify the homotopy types in all cases where linear greatest common divisor formulas for the index exist.

Finally, it follows from Joseph ([13], Theorem 8.4) that the index of a Type-B seaweed is the same as the index of a Type-C seaweed having the same flags. Consequently, the construction of Type-C meanders and related index formulas for Type-C seaweeds [10, 5], as well as results on the homotopy type of Type-C meanders, carry over Mutatas Mutandis to the Type-B case.

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