The virtual cactus group and Littelmann paths

We define a virtual cactus group and show that the cactus group action on Littelmann paths is compatible with the virtualization map defined by Pan-Scrimshaw \cite{PS18}. Our definition generalizes the group with the same name defined for the symplectic Lie algebra by the author in joint work with O. Azenhas and M. Tarighat-Feller.


Introduction
Let g be a finite dimensional, complex, semisimple Lie algebra.Let D be the Dynkin diagram associated to the root system of g, R its root system, ∆ = {α i : i ∈ D} ⊂ R the set of simple roots, W = W (R) its Weyl group, generated by the simple reflections {r i : i ∈ D}, and w 0 ∈ W the longest element of the Weyl group.For a connected sub-diagram J ⊆ D, of D, denote by θ J : J → J the unique Dynkin diagram automorphism that satisfies α θ J (j) = −w J 0 α j , for any node j ∈ J, where w J 0 is the longest element of the parabolic subgroup W J ⊆ W (the Weyl group for g restricted to J) [BB05].This leads to the following definition by Halacheva.
Definition 1. [Hal20] The cactus group J D is the group with generators s J , one for each connected subdiagram J of D, and relations given as follows: 1. s 2 J = 1; 2. s I s J = s J s I for I, J ⊆ D connected subsets if the union J ∪ I is disconnected; 3. s I s J = s θ I (J) s I if J ⊂ I.
Definition 1 is a generalization of the original definition of the cactus group defined by Henriques-Kamnitzer in [HJK04], which was denoted by J n and which corresponds to the cactus group associated to the Dynkin diagram of type A n−1 .
1.1.Main results and aim of the paper.In this paper we will be concerned with pairs of Dynkin diagrams (X, Y ) related by folding, that is, there is an injection of sets of nodes X ֒→ Y which induces an injection of the corresponding Lie algebras g X ֒→ g Y as described in [BS17].The main result and aim of this paper is the "virtualization" of the cactus group J X , as defined by Halacheva in [Hal20], and of its action on g X -crystals, transferring certain results obtained for the case C n ֒→ A 2n−1 in [ATFT22] to the more general setup described above.This is carried out in Theorem 2 and Theorem 4. It consists in defining a group monomorphism J X ֒→ J Y compatible with the action of J X and J Y on g X , respectively g Y -crystals.Moreover, by using the virtualization map on Littelmann paths described by Pan-Scrimshaw in [PS18], instead of the Baker virtualization map used in [ATFT22] for Kashiwara-Nakashima tableaux, we obtain a simple rule to compute the partial Schützenberger-Lusztig involutions of Littelmann paths in g X -crystals in terms of partial Schützenberger-Lusztig involutions of Littelmann paths in g Y -crystals.This is carried out in Theorem 4.

Acknowledgements
We thank Olga Azenhas and Mojdeh Tarighat Feller for many inspiring conversations, and ICERM for hosting the meeting titled "Research Community in Algebraic Combinatorics."We especially thank Olga Azenhas for teaching the author a great deal about the cactus group.The ideas in this paper stemmed from the wish to generalize the results in [ATFT22], which was written as part of the project titled The A, C, shifted Berenstein-Kirillov groups and cacti, in the context of the above mentioned meeting.The author was supported by ICERM for this workshop, was also supported by the grant SONATA NCN UMO-2021/43/D/ST1/02290 and partially supported by the grant MAESTRO NCN UMO-2019/34/A/ST1/00263.

The cactus group and crystals
Let Λ be the integral weight lattice and Λ + ⊂ Λ be the dominant weights.Recall that irreducible finite-dimensional representations of g are in one-to-one correspondence with the set of highest weights Λ + .We now recall the definition of a semi-normal crystal as in [BS17].
Definition 2. A semi-normal g-crystal consists of a non-empty set B together with maps To each such crystal B is associated a crystal graph, a coloured directed graph with vertex set B and edges coloured by elements i ∈ D, where if f i (b) = b ′ there is an arrow b i → b ′ .We say that a crystal is irreducible if its corresponding crystal graph is connected and finite.
The finite irreducible semi-normal g-crystals are labeled by the dominant weights Λ + .Given a highest weight λ ∈ Λ + , the corresponding irreducible crystal is usually denoted by B(λ).It encodes important information about the corresponding irreducible finite dimensional representation of g, V (λ).For instance, dim(V(λ)) equals the cardinality of B, and, in the weight decomposition equals the cardinality of the set of b ∈ B(λ) such that wt(b) = µ.Moreover, for a subinterval J ⊂ D, the crystal corresponding to the Levi restriction of V (λ) corresponds to the g J -crystal B(λ) J with crystal graph obtained from the graph for B(λ) by deleting edges with labels i / ∈ J.In this paper, we will only deal with crystals whose crystal graphs decompose into connected components, each of which is isomorphic to crystals of the form B(λ).These are also known in the literature as normal crystals.

Schützenberger-Lusztig involutions.
There is an elegant internal action of the cactus group J g on crystals via partial Schützenberger-Lusztig involutions, which are generalizations of Schützenberger-Lusztig involutions originally studied by Berenstein-Kirillov and generalized by Halacheva.For a subinterval J ⊂ D, the partial Schützenberger-Lusztig involution is defined as follows on B(λ).Let v ∈ B(λ) J be a highest weight element, and let v w J 0 ∈ B(λ) J be a lowest weight element.In particular wt(v Then the partial Schützenberger-Lusztig involution is the unique involution ξ J : B(λ) → B(λ) which satisfies for each j ∈ J: ξ J is known as the Schützenberger-Lusztig involution, and denoted simply by ξ.Each partial Schützenberger-Lusztig involution acts as the corresponding Schützenberger-Lusztig involution applied to each connected component of the Levi-branched crystal B(λ) J .If our normal crystal B is not connected, partial Schützenberger-Lusztig involutions are defined in the same way as above, on each connected component.
Theorem 1 (Halacheva,[Hal20]).Let B be a normal g-crystal.The cactus group J g acts on B via partial Schützenberger-Lusztig involutions, that is, for J ⊂ D a subinterval, the assignment s J → ξ J induces a group action.

The virtual cactus group
Let X ֒→ Y be an embedding of a twisted Dynkin diagram X into a simplylaced Dynkin diagram Y given by folding.More precisely, there is a Dynkin diagram automorphism aut : Y → Y of Y such that there is an edge-preserving bijection σ : X → Y / aut.The injection of Dynkin diagrams is reflected on the Lie algebras as follows.Let g X , respectively g Y be the complex simple Lie algebras with Dynkin diagram X, respectively Y .Then the Dynkin diagram automorphism aut induces a Lie algebra automorphism aut : g Y → g Y .The set of fixed points under this automorphism has the structure of a Lie algebra isomorphic to g X [Kac90].This induces an injection g X ֒→ g Y .Below we list all such pairs, together with the values of θ X and θ Y .We use the numbering of the vertices given by [BS17].
We have aut = θ Y , except for the cases where Y = D 2n , where We proceed to define a group monomorphism J X ֒→ J Y .Its image will be isomorphic to what we call the virtual cactus group, generalizing the concept of the virtual symplectic cactus group defined in [ATFT22] for X = C n and Y = A 2n−1 .We start by stating the following lemma, which immediately follows from the description in the previous section.We will abuse notation and consider the coset σ(I) ∈ Y / aut, as a subset of Y , for I ⊂ X.Each non-simply laced Dynkin diagram we consider has what we will call in this note a branching point x 0 ∈ X, described in the table below.
For the comfort of the reader we include the corresponding Dynkin diagrams as well below.
We now consider the following elements: where s Y Ĩ are the generators of the cactus group J Y and the product is taken over the connected components Ĩ of σ(I).Our aim for the rest of this section is to prove the following result.
Theorem 2. The map defined by is a monomorphism of groups.
Lemma 1.Let I, J ⊂ X such that J ⊂ I. Then sI sJ = sθ I (J) sI Proof.First assume that θ Y = Id.This means Y = D 2n for some n ≥ 2. If I = X then σ(I) = Y , therefore the statement of Lemma 1 follows from θ Y = Id and the defining Relation 3 for the cactus group J Y .If I ⊂ X does not contain the branching point x 0 then σ| I : I → Ĩ = σ(I) is an isomorphism, hence the statement follows trivially.If I is not X but contains the branching point, then either I is of type A, σ(I) = Ĩ is of type A and σ| I : I → Ĩ is an isomorphism, which implies the claim as in the previous case, or I is of type G 2 , in which case the claim also follows easily since J is forced to consist of just one vertex.
Assume next that θ Y = aut.If I ⊂ X contains the branching point x 0 , then θ I = Id I and σ(I) = Ĩ is connected.Let us then assume first that x 0 ∈ I. Now, if x 0 ∈ J also, then σ(J) = J is connected and θ Ĩ ( J) = J.Now, if J ⊂ I does not contain a branching point but I does, then either • σ(J) = J1 ⊔ J2 has two isomorphic connected components, in which case θ Ĩ ( J1 ) = J2 and θ Ĩ ( J2 )) = J1 , or • σ(J) = J is connected and isomorphic to J, in which case θ Ĩ ( J) = J.
We conclude then that if x 0 ∈ I and σ(J) = J is connected, then as desired.Now, if x 0 ∈ I and σ J = J1 ⊔ J2 , then we still have θ I = Id, so θ I (J) = J.
We have in this case sI .This concludes the proof in the case x 0 ∈ I. Now let us assume that x 0 / ∈ I.We have two cases: The case where σ(I) is connected is trivial because since θ Y = aut, we conclude that necessarily θ σ(I) = aut | σ(I) = Id σ(I) , also σ(J) ⊂ σ(I) is connected for each J ⊂ I, and sJ = s Y σ(J) for each J ⊂ I.It remains to consider the case where σ(I) has two connected components σ(I) = Ĩ1 ⊔ Ĩ2 .It follows that for each J ⊂ I we have a decomposition into connected components σ(J) = J1 ⊔ J2 , where Ji ⊂ Ĩi , i = 1, 2. The following identity holds by case-by-case analysis: (1) Therefore we have in this case: where the last equality follows from (1).This concludes the proof in the cases where θ Y = aut and therefore the whole proof.
Definition 3. The virtual cactus group J v X is defined by generators s σ(I) , for each I ⊂ X connected subdiagram, and by the relations: 1.
It is clear from the definition that the virtual cactus group J v X is isomorphic to the cactus group J X .
Proof of Theorem 2. To show that Φ is a group morphism, we need to show three relations: (1) s2 I = Id, (2) sI sJ = sJ sI , (3) sI sJ = sθ I (J) sI .Note that the third relation has already been established in Lemma 1.To prove (1), note that since the connected components of σ(I) are disjoint, the commutation relation 2. in Definition 1 implies To show the second relation, let I, J ⊂ X be two disjoint, connected intervals.Then necessarily σ(I) and σ(J) are mutually disjoint.We have then where the third equality follows from relation 2. for J Y .Note that the image Φ(J X ) is a group isomorphic to the virtual cactus group JX via the isomorphism sI → s σ(I) .Note that this map is well defined because σ(I) = σ(J) ⇐⇒ I = J.

Virtualization of the action of the cactus group on crystals of Littelmann paths
In this section we will borrow most of our notation from [PS18] for practical purposes as well as for the comfort of the reader.Let λ ∈ Λ + .We consider P(λ) to be the Littelmann path model for λ with paths π : [0, 1] → Λ R of the form where H i,π (t) = t, α ∨ i and where Λ i ∈ Λ + are the fundamental weights for i ∈ D. The set P(λ) has the structure of a crystal isomorphic to B(λ) with weight map wt(π) = π(1).We refer the reader to [PS18] for the definition of the crystal structure using the notation we use in this section.The original and standard reference of the topic is the paper [Lit95] by Littelmann.
Recall that in this paper we consider embeddings X ֒→ Y given by folding.Let Λ X and Λ Y be the corresponding integral weight lattices.The bijection σ : X → Y / aut induces a map Ψ : Λ X → Λ Y given by the assignment where γ i is given by Table 5.1 in [BS17] and where Λ X i and Λ Y j denote the fundamental weights in Λ X , respectively Λ Y .Definition 4. Let B be a normal g Y -crystal, and a subset V ⊂ B. The virtual root operators of type X are, for i ∈ X: where ẽi , fi , i ∈ Y are the root operators for the g Y -crystal B.
A virtual crystal is a pair (V, B) such that V has a g X -crystal structure defined by where εj , φj j ∈ Y denote the maps given by εi One of the main results in [PS18] is the following theorem.
Theorem 3 (Pan-Scrimshaw, [PS18]).The assignment π → Ψ(π) induces a virtualization map The principal aim of this section is to describe the action of the cactus group in terms of the virtualization map of Pan-Scrimshaw.For this, given a connected subdiagram I ⊂ X, let ξσ(I) := ξ Y Ĩ where ξ Y Ĩ are the partial Schützenberger-Lusztig involutions in P(ψ(λ)) and the product is taken over the connected components Ĩ of σ(I).Our next aim is to prove the following result, which generalizes [ATFT22, Theorem 5, Theorem 6, Section 9.5]. .
Proof.First note that since the Littelmann path model P(ψ(λ)) is stable under the root operators ẽi , fi , it is also stable under the action of the operators ξY σ(I) for I ⊂ X connected.Therefore, all paths in ξY σ(I) (Ψ(P(λ))) must be of the form (6), so the left inverse Ψ −1 can be explicitly computed on ξY σ(I) (Ψ(P(λ))), simply by writing out the corresponding path in this form.We now proceed to show that the diagram commutes.Let where the product is taken over the connected components Ĩ of σ(I).To continue our computations we consider three cases separately: (1) The subdiagram σ(I) = Ĩ ⊂ Y is connected.Then θ I = Id, we have γ i j = 1 if and only if σ(i j ) = ĩ1 j , ĩ2 j or σ(i j ) = ĩ1 j , ĩ2 j , ĩ3 j and γ i j = 2, 3 if and only if σ(i j ) = ĩj .In case γ i j = 1 we have θ Ĩ ( ĩ1 j ) = ĩ2 j and θ Ĩ ( ĩ2 j ) = ĩ1 j .Moreover, the root operators ẽĩ1 j and ẽĩ2 j commute.In case γ i j = 2, 3 we have θ Ĩ ( ĩj ) = ĩj .All together this implies: