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Kam10 , m OSS03c rsa [ crystal utemr,Kashiwara Furthermore, . mfligi virtual- a is folding am Kas96 Ste03 Kam07 , KN94 ia–ehdi(LS) mibai–Seshadri , o similarity. for ] Kas90 ,wihdetermines which ], ua virtualization tural SS15b o tde.Vir- studied. ion , , Kam10 LP08 , U ls ,weethe where ], Kas91 q ( B b g i , )-crystal salso is , admits ] Lit95b Lit95b Bak00 ] B ( λ ], ) , , We remark that this work partially overlaps with the work of Naito and Sagaki on LS paths invariant under a diagram automorphism [NS01, NS05a, NS10]. In their case, they only consider the fixed point subalgebras and allow automorphisms where there may be adjacent vertices in an orbit (e.g., the middle edge in the natural type A2n folding). Whereas in our case, we allow for arbitrary scaling factors and the vertices in the orbits of our automorphisms must not be adjacent. (1) (2) (2)† (2) For example, this allows us to consider types Dn+1, A2n , A2n as foldings of type A2n−1. The main application of our construction is about a particular class of finite crystals of affine type called Kirillov–Reshetikhin (KR) crystals [FOS09]. KR crystals and their associated KR modules are known to have many fascinating properties. KR modules are classified by their Drinfel’d polynomials and correspond to the minimial affinization of B(rΛr) [Cha95, CP95a, CP95b, CP96a, CP96b, CP98]. Their characters (resp. q-characters [Nak03]) satisfy the Q-system (resp. T-system) relations [Cha01, Her10]. The graded characters of (resp. Demazure submodules of) tensor products of KR modules are (resp. Nonsymmetric) Macdonald polynomials at t = 0 [LNS+15, LNS+16] (resp. [LNS+17]). KR modules conjecturally admit crystal bases [HKO+99, HKO+02] and is known for most cases [JS10, KMOY07, LNS+15, LNS+16, OS08, Yam98]. KR crystals are (conjecturally) generally perfect [FOS10, KMOY07, Yam98], a technical property that allows them to be used to construct the Kyoto path model [KKM+92], and known to be simple crystals in non-exceptional types [Oka13]. KR crystals are also used to construct soliton cellular automata, where we refer the reader to [IKT12] and references therein. Tensor products of KR crystals are also (conjecturally) in bijection with rigged configurations, which arose from the study of the Bethe Ansatz of Heisenberg spin chains [KKR86, KR86, OSS13, OSSS17, OSS03a, OSS03b, OSS03c, SS15b, SS06, Scr16]. N ri,1 Tensor products of KR crystals of the form i=1 B were constructed by Naito and Sagaki using projected level-zero LS paths in [NS03, NS06, NS08a, NS08b] and using quantum LS paths and the quantum alcove model in [LNS+15, LNSN+16]. It was conjectured that KR crystals are also well-behaved under virtualization [OSS03c] and has been proven in a variety of special cases in a type-dependent fashion [Oka13, OSS03b, OSS03c, SS15b]. We use the projected level-zero construction of Naito and Sagaki to give partial uniform results of the aforementioned conjecture. This note is organized as follows. In Section 2, we give the necessary background on crystals, the Littelmann path model, level-zero crystals, and Kirillov–Reshetikhin crystals. In Section 3, we prove our main result, that the map on weight lattices naturally induces a virtualization map. In Section 4, we apply our main result to show special cases of the conjectural virtualization of KR crystals.

Acknowledgements We would like to thank Ben Salisbury for useful comments on an early draft of this paper. We thank the anonymous referees for comments. This work benefited from computations in SageMath [SCc15, Sag17].

2. Background Let g be a (symmetrizable) Kac–Moody algebra with index set I, (generalized) Cartan matrix A = (Aij )i,j∈I, fundamental weights {Λi | i ∈ I}, simple roots {αi | i ∈ I}, simple coroots ∨ ∨ ∨ {αi | i ∈ I}, weight lattice P , coweight lattice P , root lattice Q, and coroot lattice Q . Let ∗ ∨ Uq(g) be the corresponding quantum group, hR = R ⊗Z P , and hR = R ⊗Z P be the corresponding ∗ dual space. Let h·, ·i : hR × hR → R denote the canonical pairing by evaluation, in particular ∨ ∨ hαi , αji = Aij and hαi , Λji = δij, where δij is the Kronecker delta. Let {si | i ∈ I} denote the set ∨ + of simple reflections on P , where si(λ)= λ − hαi , λi αi. Let P denote the positive weight lattice, i.e., all nonnegative linear combinations of the fundamental weights. 2 2.1. Crystals. An abstract Uq(g)-crystal is a non-empty set B along with maps

ei,fi : B → B ⊔ {0}, wt: B → P, which satisfy the conditions: ∨ (1) ϕi(b)= εi(b)+ hαi , wt(b)i, ′ ′ ′ (2) fib = b if and only if b = eib for b ∈ B, for all b ∈ B and i ∈ I, where Z k εi(b) = max{k ∈ | ei b 6= 0}, Z k ϕi(b) = max{k ∈ | fi b 6= 0}.

The maps ei and fi are known as the crystal operators. We are restricting ourselves to regular crystals in this note, and so our definition is less general than than the usual definition of crystals (see, e.g., [Kas90, Kas91]). Therefore the entire i-string through an element b ∈ B can be given diagrammatically as

εi(b) i i i i i i ϕi(b) ei b −−−−→· · · −−−−→ eib −−−−→ b −−−−→ fib −−−−→· · · −−−−→ fi b.

Let B1 and B2 be abstract Uq(g)-crystals. A crystal morphism ψ : B1 → B2 is a map B1 ⊔{0}→ B2 ⊔ {0} such that (1) ψ(0) = 0; (2) if b ∈ B1 and ψ(b) ∈ B2, then wt ψ(b) = wt(b), εi ψ(b) = εi(b), and ϕi ψ(b) = ϕi(b) for all i ∈ I; ′ ′ ′


′ ′ (3) for b, b ∈ B1, fib = b , and ψ(b), ψ(b ) ∈ B2, we have ψ(fib)= fiψ(b) and ψ(eib )= eiψ(b ) for all i ∈ I. ′ ∗ 2.2. Littelmann path model. Let π,π : [0, 1] → hR, and define an equivalence relation ∼ by saying π ∼ π′ if there exists a piecewise-linear, nondecreasing, surjective, continuous function ′ ∗ φ: [0, 1] → [0, 1] such that π = π ◦φ. Let Π be the set of all piecewise-linear functions π : [0, 1] → hR such that π(0) = 0 modulo ∼. We call the elements of Π paths. Let π1,π2 ∈ Π. Let π = π1 ∗ π2 denote the concatenation of π1 by π2: π (2t) 0 ≤ t ≤ 1/2, π(t) := 1 (π1(1) + π2(2t − 1) 1/2

π(t)= Hi,π(t)Λi. (2.1) ∈ Xi I Let mi(π) := min{Hi,π(t) | t ∈ [0, 1]} denote the minimal value of Hi,π. (k) Z We want to define ei , where k ∈ >0. If mi(π) ≤ −k, then fix t1 ∈ [0, 1] minimal such that Hi,π(t1) = mi(π) and let t0 ∈ [0, 1] maximal such that Hi,π(t) ≥ mi(π)+ k for t ∈ [0,t0]. Choose t0 = t(0)

(1) Hi,π(t(j−1))= Hi,π(t(j)) and Hi,π(t) ≥ Hi,π(t(j−1)) for t ∈ [t(j−1),t(j)]; (2) or Hi,π is strictly decreasing on [t(j−1),t(j)] and Hi,π(t) ≥ Hi,π(t(j)) for t ≤ t(j−1). 3 Set t(−1) := 0 and t(r+1) := 1, and denote by π(j) the path defined by

π(j)(t) := π t(j−1) + t (t(j) − t(j−1)) − π(t(j−1)) for i = 0, 1, . . . , r + 1.
Z (k) Definition 2.1. Fix some k ∈ >0. If mi(π) > −k, then ei π = 0. Otherwise, (k) ei π = π(0) ∗ η(1) ∗ η(2) ∗··· η(r) ∗ π(r+1), where η(j) = π(j) if Hi,π behaves as in (1) and η(j) = si(π(j)) if Hi,π behaves as in (2) on [t(j−1),t(j)]. (k) Z Next we want to define fi , where k ∈ >0. Let t0 ∈ [0, 1] be maximal such that Hi,π(t0) = mi(π). If Hi,π(1) − mi(π) ≥ k, then fix t1 ∈ [t0, 1] minimal such that Hi,π(t) ≥ mi(π)+ k for t ∈ [t1, 1]. Choose t0 = t(0) < t(1) < · · · < t(r) = t1 such that either

(1) Hi,π(t(j−1))= Hi,π(t(j)) and Hi,π(t) ≥ Hi,π(t(j−1)) for t ∈ [t(j−1), t(j)]; or (2) Hi,π is strictly increasing on [t(j−1), t(j)] and Hi,π(t) ≥ Hi,π(t(j)) for t ≥ t(j).

Let t(−1) := 0 and t(r+1) := 1, and denote by π(j) the path defined by

π(j)(t) := π t(j−1) + t (t(j) − t(j−1)) − π(t(j−1)) for i = 0, 1, . . . , r + 1. It is clear thatπ = π(0) ∗ π(1) ∗···∗ π(
r+1). Z (k) Definition 2.2. Fix some k ∈ >0. If Hi,π(1) − mi(π) < k, then fi π = 0. Otherwise, (k) fi π = π(0) ∗ η(1) ∗ η(2) ∗··· η(r) ∗ π(r+1), where η(j) = π(j) if Hi,π behaves as in (1) and η(j) = si(π(j)) if Hi,π behaves as in (2) on [t(j−1), t(j)].

Example 2.3. Consider g of type C2 and the highest weight vector π ∈ B(3Λ1 + Λ2). Thus we have

α2

f1π

Λ2 π 0 Λ1

α1 f2π

(1) (1) Remark 2.4. If k = 1, we will simply write ei and fi for ei and fi respectively. Lemma 2.5 ([Kas96]). We have (k) k (k) k ei = ei and fi = fi for all i ∈ I and k ∈ Z>0. Note that (k) (k) ∨ ∨ fi (π)= ei (π ) . (2.2) 4
Theorem 2.6 ([Kas96, Lit95a, Lit95b]). Let B(π) denote the closure of π under ei and fi, for all i ∈ I. The set B(π) is a Uq(g)-crystal. Moreover, if π is a path in the dominant chamber (i.e., + Hi,π(t) ≥ 0 for all i ∈ I and t ∈ [0, 1]) such that π(1) = λ ∈ P , then B(π) is isomorphic to the highest weight crystal of weight λ. For some path η, we say B(η) has the integrality property if for each π ∈ B(η) and i ∈ I, the values Hi,π(1) and all local minimums of Hi,π(t) are integers. We will also need the following fact. Lemma 2.7 ([Lit95b, Corollary 1]). Suppose η denote a path in the dominant chamber. Then B(η) has the integrality property. Let B(λ) denote B(π), where π is the straight line path from 0 to λ, explicitly given by π(t)= tλ. If λ ∈ P +, then B(λ) is the set of all LS paths.

2.3. Virtualization maps. Let A and A be Cartan matrices with indexing sets I and I respec- tively. We call a map φ: I → I a generalized diagram folding if for all j 6= j′, we have φ(j)= φ(j′) implies Aj,j′ = 0 (i.e., the nodes are non-adjacent).b Consider a generalized diagram foldingb φ and Z b ∗ ∗ scaling factors (γi ∈ >0)i∈Ib. Let Ψ: hR → hR be the map of the corresponding weight spaces given by b

Λi 7→b γjΛj. (2.3) −1 j∈Xφ (i) b Definition 2.8. Let B be a Uq(g)-crystal and V ⊆ B. Let φ and (γa ∈ Z>0)a∈I be the generalized diagram folding and the scaling factors. The virtual crystal operators (of type g) are defined as

b b v b γj ei = ej , (2.4a) −1 j∈φY(i) v bγj fi = fj . (2.4b) −1 j∈φY(i) b A virtual crystal is a pair (V, B) such that V has a Uq(g)-crystal structure defined by v v ei := e fi := f , b i i −1 −1 εi := γj εj ϕi := γj ϕj, (2.5) wt := Ψ−1 ◦ wt. b b for any j ∈ φ−1(i). c Remark 2.9. We note that because the nodes in φ−1(i) are non-adjacent, Equation (2.4) is well- defined since all of the crystal operators of type g used commute.

Suppose B is isomorphic to the virtual crystal (V, B) (as Uq(g)-crystals), then we call the iso- morphism a virtualization map. b b 2.4. Level-zero and Kirillov–Reshetikhin crystals. In this section, we describe two classes of crystals of affine type that will be used in Section 4: level-zero crystals and Kirillov–Reshetikhin crystals. For this section, let g be of affine type. For type g, let 0 ∈ I denote the special node, and let I0 := I \ {0} be the index set of the ∨ ∨ corresponding classical type g0. Let δ = i∈I aiαi and c = i∈I ai αi denote the null root and ∨ the canonical central element of g respectively, where ai and ai are the Kac and dual Kac labels, respectively, as given in [Kac90, Table Aff.P 1]. P 5 We say a weight λ ∈ P is a level-zero weight if λ(c) = 0. A level-zero weight is level-zero ∨ dominant if hλ, αi i ≥ 0 for all i ∈ I0. The level-zero fundamental weights {̟i ∈ P | i ∈ I0} are defined as ∨ ̟i = Λi − ai Λ0. (2.6) ′ Let Uq(g) := Uq([g, g]) be the quantum group corresponding to the derived subalgebra of g. ∗ ∗ Define the classical projection as cl: hR → hR/Rδ as the canonical projection. We identity the ′ weight lattice of Uq(g) with Pcl := {cl(λ) | λ ∈ P }. ′ We now recall an important class of finite-dimensional Uq(g)-modules called Kirillov–Reshetikhin r,s (KR) modules denoted by W , where r ∈ I0 and s ∈ Z>0, which are classified by their Drinfeld polynomials [CP95b, CP98]. KR modules conjecturally admit a crystal bases [HKO+99, HKO+02], which has been shown to exist for all non-exceptional types in [OS08] and in certain cases in exceptional types [JS10, KMOY07, LNS+15, LNS+16, Yam98]. The corresponding crystal to W r,s is known as a Kirillov–Reshetikhin (KR) crystal and is denoted by Br,s. KR crystals are also conjecturally well-behaved under the natural virtualization induced from the diagram folding φ given by the well-known natural embeddings of algebras [JM85]:

(1) (2) (2)† (2) (1) ,Cn , A2n , A2n , Dn+1 ֒−→ A2n−1 (1) (2) (1) , Bn , A n− ֒−→ Dn 2 1 +1 (2.7) (2) (1) (1) , E6 , F4 ֒−→ E6 (1) (3) (1) . G2 , D4 ֒−→ D4

−1 In this case, we define the scaling factors (γj)j∈Ib by γj = γi for all j ∈ φ (i), where γi is given as follows. (1) Suppose the Dynkin diagram of g has a unique arrow. (a) Suppose the arrow points towards the component of the special node 0. Then γi = 1 for all i ∈ I. (b) Otherwise, γi is the order of φ for all i in the component of 0 after removing the arrow and γi = 1 in all other components. (1) (2) Otherwise the Dynkin diagram of g has 2 arrows and is a folding of type A2n−1. Then γi = 1 for all 1 ≤ i ≤ n − 1, and for i ∈ {0,n}, we have γi = 2 if the arrow points away from i and γi = 1 otherwise. Conjecture 2.10. [OSS03c, Conj. 3.7] The KR crystal Br,s of type g virtualizes into

n,s n,s (2) (2)† r,s B ⊗ B if g = A2n , A2n and r = n, B = ′ r ,γrs ( r′∈φ−1(r) B otherwise.

b N r,1 (2) (2) (1) (2) Conjecture 2.10 was shown for B in types Dn+1, A2n , and Cn in [OSS03b] and types E6 , (1) (1) (3) 1,s F4 (except r = 2), G2 , and D4 in [SS15b], B for all non-exceptional types [OSS03c], and r,s (1) (2) B for r

6 Λ1 − Λ0 − δ Λ1 − Λ0 −Λ0 + Λ1

1 1 1

Λ2 − Λ1 − δ Λ2 − Λ1 −Λ1 + Λ2

2 0 2 2 0

Λ1 − Λ2 − δ Λ1 − Λ2 Λ1 − Λ2

1 1 1

Λ0 − Λ1 − δ Λ0 − Λ1 Λ0 − Λ1

1,1 Figure 1. A portion of the level-zero crystal B(Λ1 −Λ0) (left) and B constructed (1) from Υ (right) in type C2 .

Theorem 2.11. Let g be of affine type. Let λ = m ̟ , with m ∈ Z , is a dominant i∈I0 i i i ≥0 level-zero weight. Then there exists a canonical U ′ (g)-crystal isomorphism q P i,1 ⊗mi Υ: B(λ)cl → B . i∈I O0
1,1 Example 2.12. The construction of B from the classical projection of B(Λ1 − Λ0) is given in Figure 1.

3. Main results In this section, we prove our main result. Theorem 3.1. Fix a map φ: I → I. The induced map Ψ: P → P given by Equation (2.3) induces a virtualization map Ψ: B(λ) → B Ψ(λ) given by b b
e Ψ(π)(t)= Hi,π(t)Ψ(Λi) ∈ Xi I if and only if the following propertiese hold: ′ −1 (I) For all j 6= j ∈ φ (i), we have Aj,j′ = 0 (i.e., φ is a generalized diagram folding). (II) We have b Ψ(αi)= γjαj. (3.1) −1 j∈Xφ (i) Proof. Fix some π ∈ B(λ), and set b

π(t) := Ψ(π)(t)= Hi,π(t) γjΛj. ∈ −1 Xi I j∈Xφ (i) b e 7 b ∨ ∨ v It is clear that Ψ(π ) = Ψ(π) . So Ψ is a virtualization map if and only if Ψ(eiπ) = ei Ψ(π) by Equation (2.2), where Equation (2.2) also holds for the virtual crystal operators. Note that
′ −1 εi(π) = − mint He i,π(t), ande hence, wee have εj(π)/γj = εj′ (π)/γj′ = εi(π) for alle j, j ∈ φ e(i). Thus note that property (I) holds if and only if the virtual crystal operators are well-defined (see Remark 2.9). b b b b Fix some i ∈ I. We first consider the case when eiπ = 0. Thus we have mi(π) > −1. By −1 Lemma 2.7, we can assume mi(π) ≥ 0. Next, take any k ∈ φ (i). Then

mk(π) = min{Hk,πb(t) | t ∈ [0, 1]} ∨ = min{ αk , π(t) | t ∈ [0, 1]} b = min{γ kHi,π(t) | t ∈ [0, 1]} b b = γkmi(π) ≥ 0.

−1 Therefore ekπ = 0 for any k ∈ φ (i). −1 Now we consider the case when eiπ 6= 0. Thus, for any j ∈ φ (i), we have

b b ∨ Hj,πb(t)= αj , Ψ(π)(t) = γjHi,π(t). D E Fix some division of π into subpaths b e

eiπ = π(0) ∗ η(1) ∗ η(2) ∗···∗ η(r) ∗ π(r+1) as in Definition 2.1. Note that

Ψ(π)= Ψ(π(0)) ∗ Ψ(π(1)) ∗ Ψ(π(2)) ∗··· Ψ(π(r)) ∗ Ψ(π(r+1)) since conditions (1)e and (2)e for the subdivisione e of π still holde wheneHi,π is scaled by a positive constant. Thus we have

Ψ(eiπ)= Ψ(π(0)) ∗ Ψ(η(1)) ∗ Ψ(η(2)) ∗···∗ Ψ(η(r)) ∗ Ψ(π(r+1)), and to show e e e e e e v ei Ψ(π)= π(0) ∗ η(1) ∗ η(2) ∗···∗ η(r) ∗ π(r+1) = Ψ(eiπ), it is sufficient to show that η = Ψ(η ) for all 1 ≤ k ≤ r since π = Ψ(π ) and π = e (k)b b (k) b b b e(0) (0) (r+1) Ψ(π(r+1)). Fix some 1 ≤ k ≤ r. If η(kb) = π(k)e, then Hi,π(t(k−1)) = Hi,π(t(k)) andb Hi,πe(t) ≥ Hi,π(t(bk−1)) for −1 te∈ [t(k−1),t(k)], i.e., condition (1) is satisfied. So for any j ∈ φ (i) and t ∈ [t(k−1),t(k)], we obtain

Hj,πb(t(k))= γjHi,π(t(k))= γjHi,π(t(k−1))= Hj,πb(t(k−1)), and

Hj,πb(t)= γjHi,π(t) ≥ γjHi,π(t(k−1))= Hj,πb(t(k−1)) since γi is positive. Therefore η(k) = Ψ(π(k))= Ψ(η(k)). Now suppose η(k) = siπ(k), which implies Hi,π is strictly decreasing on [t(k−1),t(k)] and Hi,π(t) ≥ Hi,π(t(k)) for t ≤ t(k−1), i.e., conditionb e (2) is satisfied.e Then ∨ η(k)(t) = (siπ(k))(t)= si π(k)(t) = π(k)(t) − αi ,π(k)(t) αi = π(k)(t) − Hi,π(t)αi, and

Ψ(η(k))(t)=Ψ(π(k)(t) − Hi,π(t)αi)=Ψ π(k)(t) − Hi,π(t)Ψ(αi). 8
e We note that from the definition of the crystal operators and Lemma 2.5, it is sufficient to v v consider si = j∈φ−1(i) sj for the action of ei . Thus we have

Qv si Ψ π(k)(bt) = sj Ψ π(k)(t) j∈φ−1(i)

Y

b = s H (t) γ Λ j  l,π p p −1 ∈ −1 j∈φY(i) Xl I p∈Xφ (l) b  b  ∨ = Ψ(π(k)(t)) − αj , Hl,π(t) γpΛp αj −1 * −1 + j∈Xφ (i) Xl∈I p∈Xφ (l) b b b

=Ψ π(k)(t) − Hi,π(t) γjαj. j∈φ−1(i)
X v b Hence si Ψ(π(k)) = Ψ(η(k)) if and only if Ψ(αi) = j∈φ−1(i) αj. Therefore Ψ is a virtualization map if and only if Equation (3.1) holds. 
P e e b e Remark 3.2. Theorem 3.1 holds for any path η such that B(η) (and subsequently B(η)) has the integrality property.

−1 b Example 3.3. Consider the diagram folding from type A3 to type C2, where φ (1) = {1, 3} and −1 φ (2) = {2} and γi = i. The map Ψ is given by

Λ1 7→ Λ1 + Λ3, Λ 7→ 2Λ . 2 b 2 b ′ ′ Since αi = i′∈I Ai ,iΛi , we have b P α1 + α3 = (2Λ1 − Λ2) + (2Λ3 − Λ2) = 2(Λ1 + Λ3) − 2Λ2

= 2Ψ(Λ1) − Ψ(Λ2)=Ψ(α1), b b b b b b b b b 2α2 = 2(2Λ2 − Λ1 − Λ3) = 2(2Λ2) − 2(Λ1 + Λ3)

= 2Ψ(Λ2) − 2Ψ(Λ1)=Ψ(α2). b b b b b b b Therefore, the map Ψ is a virtualization map by Theorem 3.1. In particular, if we consider the highest weight element π(t) = 3tΛ1 + tΛ2 ∈ B(3Λ1 + Λ2), then e (f2(π))(t) = 5tΛ1 − tΛ2,

Ψ(π)(t) = 3tΛ1 + 2tΛ2 + 3tΛ3. Ψ(f π)(t)= f 2Ψ(π) (t) = 5tΛ − 2tΛ + 5tΛ . e2 2 1 2 3
e b e 4. Applications In this section, we present an application of Theorem 3.1 to KR crystals.

Proposition 4.1. Let g be of affine type. Let Ψ be the virtualization map induced from the gener- ′ alized diagram folding φ given in Section 2.4. Then there exists a Uq(g)-crystal virtualization map e 9 Ψcl such that the diagram e e B(λ) Ψ / B Ψ(λ)
cl cl

  / B(λ)cl e B Ψ(λ) cl Ψcl
commutes.

Proof. Note that Ψ(δ)= a0γ0δ, and that cl is given by quotienting by δ. Define Ψcl : Pcl → Pcl by

b b Λi 7→ γi Λj, −1 j∈Xφ (i) b and it is straightforward to verify that

αi 7→ γi αj. −1 j∈Xφ (i) b Recall that the computation of the crystal operators does not depend on the coefficient of δ. Hence,  we define Ψcl : B(λ)cl → B Ψ(λ) cl as the crystal map induced from Ψcl, and the claim follows.
e (2) (2)† Theorem 4.2. Let g be of affine type. Suppose r ∈ I is such that γr = 1 or g is of type A2n , A2n . Then Conjecture 2.10 holds for s = 1.

Proof. From Theorem 3.1, there exists a virtualization map Ψ: B(̟r) → B Ψ(̟r) from the diagram folding φ given in Section 2.4. From Proposition 4.1 and Theorem 2.11, the result follows.
e 

Note that we require γr =1 as

′ ∼ r ,1 ⊗γr B Ψ(̟r) cl = (B ) , r′∈φ−1(r)
O which does not agree with Conjecture 2.10 when γr > 1.

(1) 1,1 Example 4.3. Consider type C2 and B from Example 2.12. Recall that for the folding of (1) (1) type A3 onto C2 , we have γ0 = γ2 = 1 and γ1. Applying Υ to the image of B(Λ1 − Λ0) in 10 B(Λ1 + Λ3 − 2Λ0) results in the virtual crystal

b b b Λ1 + Λ3 − 2Λ0

1

2Λ2 − Λ1 − Λ3

2 0

Λ1 + Λ3 − 2Λ2

1

2Λ0 − Λ1 − Λ3

Appendix A. Examples with Sage

We give an example of the virtualization map of B(Λ1) of type C2 into B(Λ1 + Λ3) of type A3 using Sage [Sag17]. b sage : PC = RootSystem([ ’C’ ,2]).weight_space() sage : LaC = PC.fundamental_weights() sage : BC = crystals.LSPaths(LaC[1]) sage : PA = RootSystem([ ’A’ ,3]).weight_space() sage : LaA = PA.fundamental_weights() sage : BA = crystals.LSPaths(LaA[1]+LaA[3]) sage : gens = BA.module_generators sage : psi = BC.crystal_morphism(gens, codomain=BA) sage : for x in BC : ....: print "C2:" , x ....: print "A3:" , psi(x) ....: C2: (Lambda[1],) A3: (Lambda[1] + Lambda[3],) C2: (-Lambda[1] + Lambda[2],) A3: (-Lambda[1] + 2*Lambda[2] - Lambda[3],) C2: (Lambda[1] - Lambda[2],) A3: (Lambda[1] - 2*Lambda[2] + Lambda[3],) C2: (-Lambda[1],) A3: (-Lambda[1] - Lambda[3],)

Next we explicitly show the virtual crystal operators act as desired.

sage : mg = BA.highest_weight_vector(); mg (Lambda[1] + Lambda[3],) sage : x1 = mg.f_string([1,3]); x1 (-Lambda[1] + 2*Lambda[2] - Lambda[3],) sage : x2 = x1.f_string([2,2]); x2 (Lambda[1] - 2*Lambda[2] + Lambda[3],) sage : x3 = x2.f_string([1,3]); x3 (-Lambda[1] - Lambda[3],)

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(J. Pan) Department of Mathematics, University of California, One Shields Avenue, Davis, CA 95616 E-mail address: [email protected]

(T. Scrimshaw) School of Mathematics, University of Minnesota, 200 Oak St. SE, Minneapolis, MN 55455 E-mail address: [email protected] URL: https://sites.google.com/view/tscrim/home

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