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Perfectness of Kirillov--Reshetikhin crystals for nonexceptional types.

1 Introduction

Kirillov-Reshetikhin (KR) crystals [B.sup.r,s] are crystals corresponding to finite-dimensional [U'.sub.q](g)-modules [3], [4], where g is an affine Kac-Moody algebra. Recently, a lot of progress has been made regarding long outstanding problems concerning these crystals which appear in mathematical physics and the path realization of affine highest weight crystals [13]. In [20, 21] the existence of KR crystals was shown. In [5] a major step in understanding these crystals was provided by giving explicit combinatorial realizations for all nonexceptional types. This abstract is based on [5, 6]. We prove a conjecture of Hatayama, Kuniba, Okado, Takagu, and Tsuboi [8, Conjecture 2.1] about the perfectness of these KR crystals.

Conjecture 1.1 [8], Conjecture 2.1] The Kirillov-Reshetikhin crystal [B.sup.r,s] is perfect if and only if s/[c.sub.r] is an integer with [c.sub.r] as in Table 1. If [B.sup.r,s] is perfect, its level is s/[c.sub.r].

In [14], this conjecture was proven for all [B.sup.r,s] for type [A.sup.(1).sub.n], for [B.sup.1,s] for nonexceptional types (except for type [C.sup.(1).sub.n]), for [B.sup.n-1,s], [B.sup.n,s] of type [D.sup.(1).sub.n], and [B.sup.n,s] for types and [D.sup.(2).sub.n+1]. When the highest weight is given by the highest root, level-1 perfect crystals were constructed in [1]. For 1 [less than or equal to] r [less than or equal to] n - 2 for type [D.sup.(1).sub.n], 1 [less than or equal to] r [less than or equal to] n - 1 for type [B.sup.(1).sub.n], and 1 [less than or equal to] r [less than or equal to] n for type [A.sup.(2).sub.2n-1], the conjecture was proved in [22]. The case [G.sup.(1).sub.2] and r = 1 was treated in [24] and the case [D.sup.(3).sub.4] and r = 1 was treated in [16]. Naito and Sagaki [18] showed that the conjecture holds for twisted algebras, if it is true for the untwisted simply-laced cases.

In this paper we prove Conjecture 1.1 in general for nonexceptional types.

Theorem 1.2 If g is of nonexceptional type, Conjecture 1.1 is true.

The paper is organized as follows. In Section 2 we give basic notation and the definition of perfectness in Definition 2.1. In Section 3 we review the realizations of the KR crystals of nonexceptional types as recently provided in [5]. Section 4 is reserved for the proof of Theorem 1.2 and an explicit description of the minimal elements [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the perfect crystals. A long version of this article containing further details and examples is available at [6].

2 Definitions and perfectness

We follow the notation of [12, 5]. Let B be a [U'.sub.q](g)-crystal [15]. Denote by [[alpha].sub.i] and [[LAMBDA].sub.i] for i [member of] I the simple roots and fundamental weights and by c the canoncial central element associated to g, where I is the index set of the Dynkin diagram of g (see Table 2). Let P = [[direct sum].sub.i[member of]I][Z[LAMBDA].sub.i] be the weight lattice of g and [P.sup.+] the set of dominant weights. For a positive integer l, the set of level-l weights is

[P.sup.+.sub.l] = {[LAMBDA] [member of] [P.sup.+] | lev([LAMBDA]) = l}.

where lev([LAMBDA]) := [LAMBDA](c). The set of level-0 weights is denoted by [P.sub.0]. We identify dominant weights with partitions; each [[LAMBDA].sub.i] yields a column of height i (except for spin nodes). For more details, please consult [11].

We denote by [f.sub.i], [e.sub.i] : B [right arrow] B [union] {[??]} for i [member of] I the Kashiwara operators and by wt : B [right arrow] P the weight function on the crystal. For b [member of] B we define [[epsilon].sub.i](b) = max{k | [e.sup.k.sub.i](b) [not equal to] [??]}, [[phi].sub.i](b) = max{k | [f.sup.k.sub.i](b) [not equal to] [??]}, and

[epsilon](b) = [summation over (i[member of]I)] [[epsilon].sub.i](b)[[LAMBDA].sub.i] and [phi](b) = [summation over (i[member of])I] [[phi].sub.i](b)[[LAMBDA].sub.i].

Next we define perfect crystals, see for example [11].

Definition 2.1 For a positive integer l > 0, a crystal B is called perfect crystal of level l, if the following conditions are satisfied:

1. B is isomorphic to the crystal graph of a finite-dimensional [U'.sub.q](g)-module.

2. B [cross product] B is connected.

3. There exists a [lambda] [member of] [P.sub.0], such that wt (B) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and there is a unique element in B of classical weight [lambda].

4. [for all] b [member of] B, lev([epsilon](b)) [greater than or equal to] l.

5. [for all] [LAMBDA] [member of] [P.sup.+.sub.l], there exist unique elements [b.sub.[LAMBDA]], [b.sup.[LAMBDA]] [member of] B, such that

[epsilon]([b.sub.[LAMBDA]]) = [LAMBDA] = [phi]([b.sup.[LAMBDA]]).

We denote by [B.sub.min] the set of minimal elements in B, namely

[B.sub.min] = {b [member of] B | lev([epsilon](b)) = l}.

Note that condition (5) of Definition 2.1 ensures that [epsilon], [phi] : [B.sub.min] [right arrow] [P.sup.+.sub.l] are bijections. They induce an automorphism [tau] = [epsilon] o [[phi].sup.-1] on [P.sup.+.sub.l].

In [22, 5] [+ or -]-diagrams were introduced, which describe the branching [X.sub.n] [right arrow] [X.sub.n-1] where [X.sub.n] = [B.sub.n], [C.sub.n], [D.sub.n]. A [+ or -]-diagram P of shape [LAMBDA]/[lambda] is a sequence of partitions [lambda] [subset] [mu] [subset] [LAMBDA] such that LAMBDA /[mu] and [mu]/[lambda] are horizontal strips (i.e. every column contains at most one box). We depict this [+ or -]-diagram by the skew tableau of shape [LAMBDA]/[lambda] in which the cells of [mu]/[lambda] are filled with the symbol + and those of [LAMBDA]/[mu] are filled with the symbol -. There are further type specific rules which can be found in [5, Section 3.2]. There exists a bijection [PHI] between [+ or -]-diagrams and the [X.sub.n-1]-highest weight vectors inside the [X.sub.n] crystal of highest weight [LAMBDA].

3 Realization of KR-crystals

Throughout the paper we use the realization of [B.sup.r,s] as given in [5, 21, 22]. In this section we briefly recall the main constructions.

3.1 KR crystals of type [A.sup.(1).sub.n]

Let [LAMBDA] = [l.sub.0][[LAMBDA].sub.0] + [l.sub.1][[LAMBDA].sub.1] + ... + [l.sub.n][[LAMBDA].sub.n] be a dominant weight. Then the level is given by

lev([LAMBDA]) = [l.sub.0] + ... + [l.sub.n].

A combinatorial description of [B.sup.r,s] of type [A.sup.(1).sub.n] was provided by Shimozono [23]. As a {1,2, ..., n}-crystal

[B.sup.r,s] [congruent to] B(s[[LAMBDA].sub.r]).

The Dynkin diagram of [A.sup.(1).sub.n] has a cyclic automorphism [sigma](i) = i + 1 (mod n + 1) which extends to the crystal in form of the promotion operator. The action of the affine crystal operators [f.sub.0] and [e.sub.0] is given by

[f.sub.0] = [[sigma].sup.-1] o [f.sub.1] o [sigma]] and [e.sub.0] = [[sigma].sup.-1] o [e.sub.1] o [sigma].

3.2 KR crystals of type [D.sup.(1).sub.n], [B.sup.(1).sub.n], [A.sup.(2).sub.2n-1]

Let [LAMBDA] = [l.sub.0][[LAMBDA].sub.0] + [l.sub.1][[LAMBDA].sub.1] + ... + [l.sub.n][[LAMBDA].sub.n] be a dominant weight. Then the level is given by

lev([LAMBDA]) = [l.sub.0] + [l.sub.1] + 2[l.sub.2] + 2[l.sub.3] + ... + 2[l.sub.n-2] + [l.sub.n-1] + [l.sub.n] for type [D.sup.(1).sub.n] lev([LAMBDA]) = [l.sub.0] + [l.sub.1] + 2[l.sub.2] + 2[l.sub.3] + ... + 2[l.sub.n-2] + [l.sub.n-1] + [l.sub.n] for type [B.sup.(1).sub.n] lev([LAMBDA]) = [l.sub.0] + [l.sub.1] + 2[l.sub.2] + 2[l.sub.3] + ... + 2[l.sub.n-2] + [l.sub.n-1] + 2[l.sub.n] for type [A.sup.(2).sub.2n-1]. (3.1)

We have the following realization of [B.sup.r,s]. Let [X.sub.n] = [D.sub.n], [B.sub.n], [C.sub.n] be the classical subalgebra for [D.sup.(1).sub.n], [B.sup.(1).sub.n], [A.sup.(2).sub.2n-1], respectively.

Definition 3.1 Let 1 [less than or equal to] r [less than or equal to] n - 2 for type [D.sup.(1).sub.n], 1 [less than or equal to] r [less than or equal to] n - 1 for type [B.sup.(1).sub.n], and 1 [less than or equal to] r [less than or equal to] n for type [A.sup.(2).sub.2n-1]. Then [B.sup.r's] is defined as follows. As an [X.sub.n]-crystal

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (3.2)

where the sum runs over all dominant weights [LAMBDA] that can be obtained from s[[LAMBDA].sub.r] by the removal of vertical dominoes. The affine crystal operators [e.sub.0] and [f.sub.0] are defined as

[f.sub.0] = [[sigma].sup.-1] o [f.sub.1] o [sigma] and [e.sub.0] = [[sigma].sup.-1] o [e.sub.1] o [sigma], (3.3)

where [sigma] is the crystal automorphism defined in [22, Definition 4.2].

Definition 3.2 Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the [A.sup.(2).sub.2n-1]-KR crystal. Then [B.sup.n's] of type [B.sup.(1).sub.n] is defined through the unique injective map S : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [([m.sub.i]).sub.0[less than or equal to]i[less than or equal to]n] = (2,2, ..., 2,1).

In addition, the [+ or -]-diagrams of [A.sup.(2).sub.2n-1] that occur in the image are precisely those which can be obtained by doubling a [+ or -]-diagram of [B.sup.n,s] (see [5, Lemma 3.5]). S induces an embedding of dominant weights of [B.sup.(1).sub.n] into dominant weights of [A.sup.(2),.sub.2n-1], namely S([[LAMBDA].sub.i]) = [m.sub.i][[LAmDA].sub.i]. It is easy to see that for any A [member of] [P.sup.+] we have lev(S([LAMBDA])) = 2lev([LAMBDA]) using (3.1).

For the definition of [B.sup.n,s] and [B.sup.n-1,s] of type [D.sup.(1).sub.n], see for example [5, Section 6.2].

3.3 KR crystal of type [C.sup.(1).sub.n]

The level of a dominant [C.sup.(1).sub.n] weight [LAMBDA] = [l.sub.0][[LAMBDA].sub.0] + ... + [l.sub.n][[LAMBDA].sub.n] is given by

lev([LAMBDA]) = [l.sub.0] + ... + [l.sub.n].

We use the realization of [B.sup.r,s] as the fixed point set of the automorphism [sigma] [22, Definition 4.2] (see Definition 3.1) inside [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of [5, Theorem 5.7].

Definition 3.3 For 1 [less than or equal to] r < n, the KR crystal [B.sup.r's] of type [C.sup.(1).sub.n] is defined to be the fixed point set under [sigma] inside [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with the operators

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the Kashiwara operators on the right act in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Under the crystal embedding S : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Under the embedding S, the level of [LAMBDA] [member of] [P.sup.+] doubles, that is lev(S([LAMBDA])) = 2 lev([LAMBDA]).

For [B.sup.n,s] of type [C.sup.(1).sub.n] we refer to [5, Section 6.1].

3.4 KR crystals of type [A.sup.(2).sub.2n], [D.sup.(2).sub.n+1]

Let [LAMBDA] = [l.sub.0][[LMBDA].sub.0] + [l.sub.1][[LAMBDA].sub.1] + ... + [l.sub.n][[LAMBDA].sub.n] be a dominant weight. The level is given by

lev([LAMBDA]) = [l.sub.0] + 2[l.sub.1] + 2[l.sub.2] + ... + 2[l.sub.n-2] + 2[l.sub.n-1] + 2[l.sub.n] for type [A.sup.(2).sub.2n] lev([LAMBDA]) = [l.sub.0] + 2[l.sub.1] + 2[l.sub.2] + ... + 2[l.sub.n-2] + 2[l.sub.n-1] + [l.sub.n] for type [D.sup.(2).sub.n+1].

Define positive integers [m.sub.i] for i [member of] I as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.4)

Then [B.sup.r,s] can be realized as follows.

Definition 3.4 For 1 [less than or equal to] r [less than or equal to] n for g = [A.sup.(2).sub.2n], 1 [less than or equal to] r [less than or equal to] n for g = [D.sup.(2).sub.n+1] and s [less than or equal to] 1, there exists a unique injective map S : [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The [+ or -]-diagrams of [C.sup.(1).sub.n] that occur in the image of S are precisely those which can be obtained by doubling a [+ or -]-diagram of [B.sup.r,s] (see [5, Lemma 3.5]). S induces an embedding of dominant weights for [A.sup.(2).sub.2n], [D.sup.(2).sub.n+1] into dominant weights of type [C.sup.(1).sub.n], with S([LAMBDA].sub.i]) = [m.sub.i][[LAMBDA].sub.i]. This map preserves the level of a weight, that is lev(S([LAMBDA])) = lev([LAMDA]).

For the case r = n of type [D.sup.(2).sub.n+1] we refer to [5, Definition 6.2].

4 Proof of Theorem 1.2

For type [A.sup.(1).sub.n], perfectness of [B.sup.r,s] was proven in [14]. For all other types, in the case that s/[c.sub.r] is an integer, we need to show that the 5 defining conditions in Definition 2.1 are satisfied:

1. This was recently shown in [21].

2. This follows from [7, Corollary 6.1] under [7, Assumption 1]. Assumption 1 is satisfied except for type [A.sup.(2).sub.2n]: The regularity of [B.sup.r,s] is ensured by (1), the existence of an automorphism [sigma] was proven in [5, Section 7], and the unique element u [member of] [B.sup.r,s] such that [epsilon](u) = s[[LAMBDA].sub.0] and [phi](u) = s[[LAMBDA].sub.v] (where v = 1 for r odd for types [B.sup.(1).sub.n], [D.sup.(1).sub.n], [A.sup.(2).sub.2n-1], v = r for [A.sup.(1).sub.n], and v = 0 otherwise) is given by the classically highest weight element in the component B(0) for v = 0, B(s[[LAMBDA].sub.1]) for v = 1, and B(s[[LAMBDA].sub.r]) for v = r. Note that [[LAMBDA].sub.0] = [tau]([[LAMBDA].sub.v]), where [tau] = [epsilon] o [[phi].sup.-1]. For type [A.sup.(2).sub.2n], perfectness follows from [18].

3. The statement is true for [lambda] = s([[LAMBDA].sub.r] - [[LAMBDA].sub.r](c)[[LAMBDA].sub.0]), which follows from the decomposition formulas [2,9,10,19].

Conditions (4) and (5) will be shown in the following subsections using case by case considerations: Section 4.1 for type [A.sup.(1).sub.n], Sections 4.2, 4.3, and 4.4 for types [B.sup.(1).sub.n], [D.sup.(1).sub.n], [A.sup.(2).sub.2n,-1], Sections 4.5 and 4.6 for type [C.sup.(1).sub.n], Section 4.7 for type [A.sup.(2).sub.2n]), and Sections 4.8 and 4.9 for type [D.sup.(2).sub.n+1].

When s/[c.sub.r] is not an integer, we show in the subsequent sections that the minimum of the level of [epsilon](b) is the smallest integer exceeding s/[c.sub.r], and provide examples that contradict condition (5) of Definition 2.1 for each crystal, thereby proving that [B.sup.r,s] is not perfect. In the case that s/[c.sub.r] is an integer, we provide an explicit construction of the minimal elements of [B.sup.r,s].

4.1 Type [A.sup.(1).sub.n]

It was already proven in [14] that [B.sup.r,s] is perfect. We give below its associated automorphism [tau] and minimal elements. [tau] on P is defined by

[tau]([n.summation over (i=0)][k.sub.i][[LAMBDA].sub.i]) = [n.summation over (i=0)][k.sub.i][[LAMBDA].sub.i-r mod n+1].

Recall that [B.sup.r,s] is identified with the set of semistandard tableaux of r x s rectangular shape over the alphabet {1,2, ..., n + 1}. For b [member of] [B.sup.r,s] let [x.sub.ij] = [x.sub.ij](b) denote the number of letters j in the i-th row of b for 1 [less than or equal to] i [less than or equal to] r, 1 [less than or equal to] j [less than or equal to] n + 1. Set r' = n + 1 - r, then

[x.sub.ij] = 0 unless i [less than or equal to] j [less than or equal to] i + r'.

Let [LAMBDA] = [[summation].sup.n.sub.i=0] [l.sub.i][[LAMBDA].sub.i] be in [P.sup.+.sub.s], that is, [l.sub.0], [l.sub.1], ..., [l.sub.n] [member of] [Z.sub.[greater than or equal to]0], [[summation].sup.n.sub.i=0][l.sub.i] = s. Then [x.sub.ij](b) of the

element b such that [epsilon](b) = [LAMBDA] is given by

[x.sub.ii] = [l.sub.0] + [r-1.summation over ([alpha]=i)] [l.sub.[alpha]+r'], [x.sub.ij] = [l.sub.j-1] (i < j < i + r'), [x.sub.i,i+r'] = [i-1.summation over ([alpha]-0)] [l.sub.[alpha]+r'] (4.1)

for 1 [less than or equal to] i [less than or equal to] r.

4.2 Types [B.sup.(1).sub.n], [D.sup.(1).sub.n], [A.sup.(2).sub.2n-1]

Conditions (4) and (5) of Definition 2.1 for 1 [less than or equal to] r [less than or equal to] n - 2 for type [D.sup.(1).sub.n], 1 [less than or equal to] r [less than or equal to] n - 1 for type [B.sup.(1).sub.n], and 1 [less than or equal to] r [less than or equal to] n for type [A.sup.(2).sub.2n-1] were shown in [22, Section 6]. To a given fundamental weight [[LAMBDA].sub.k] a [+ or -]-diagram diagram([[LAMBDA].sub.k]) was associated. This map can be extended to any dominant weight [LAMBDA] = [l.sub.0][[LAMBDA].sub.0] + ... + [l.sub.n][[LAMBDA].sub.n] by concatenating the columns of the [+ or -]-diagrams of each piece. To every fundamental weight [[LAMBDA].sub.k] a string of operators f([[LAMBDA].sub.k]) can be associated as in [22, Section 6].

The minimal element b in [B.sup.r,s] that satisfies [epsilon](b) = [LAMBDA] can now be constructed as follows

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For [LAMBDA] = [[summation].sup.n.sub.i=0][l.sub.i][[LAMBDA].sub.i] [member of] [P.sup.+.sub.s], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4.3 Type [D.sup.(1).sub.n] for r = n - 1, n

The cases when r = n, n - 1 for type [D.sup.(1).sub.n] were treated in [14]. We refer to [14] or [6, Section 4.3] for an explicit description of the minimal elements.

The automorphism [tau] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4.4 Type [B.sup.(1).sub.n] for r = n

In this section we consider the perfectness of [B.sup.n,s] of type [B.sup.(1).sub.n].

Proposition 4.1 We have

min{lev([epsilon](b)) | b [member of] [B.sup.n,2s+1]} [greater than or equal to] s + 1, min{lev([epsilon](b)) | b [member of] [B.sup.n,2s]} [greater than or equal to] s.

Proof: Suppose, there exists an element b [member of] [B.sup.n,2s+1] with lev([epsilon](b)) = p < s + 1. Since [B.sup.n,2s+1] is embedded into [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by Definition 3.2, this would yield an element [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with lev([??]) < 2s + 1.

But this is not possible, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is a perfect crystal of level 2s + 1.

Suppose there exists an element b [member of] [B.sup.n,2s] with lev([member of](b)) = p < s. By the same argument one obtains a contradiction to the level of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Hence to show that [B.sup.n,2s+1] is not perfect, it is enough to provide two elements [b.sub.1], [b.sub.2] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] which are in the realization of [B.sup.r,s] under S and satisfy [member of]([b.sub.1]) = [epsilon]([b.sub.2]) = [LAMBDA], where lev([LAMBDA]) = 2s + 2. We use the notation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 4.2 Define the following elements [b.sub.1], [b.sub.2] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]: For n odd, let [P.sub.1] be the [+ or -]-diagram corresponding to one column of height n containing one +, and 2s columns of height 1 each containing a - sign, and [P.sub.2] the analogous [+ or -]-diagram but with a - in the column of height n. Set [??] = [(n, (n - 1).sup.2], n, [(n - 2).sup.2], [(n - 1).sup.2], n, ..., [2.sup.2], ..., [(n - 1).sup.2], n) and

[b.sub.1] = [??]([PHI]([P.sub.1])) and [b.sub.2] = [??]([PHI]([P.sub.2])).

For n even, replace the columns of height 1 with columns of height 2 and fill them with [+ or -]-pairs. Then [b.sub.1],[b.sub.2] [member of] S([B.sup.n-2s+1]) and [epsilon]([b.sub.1]) = [member of]([b.sub.2]) = 2s[[LAMBDA].sub.1] + [A.sub.n], which is of level 2s + 2.

Proof: It is clear from the construction that the [+ or -]-diagrams corresponding to [b.sub.1] and [b.sub.2] can be obtained by doubling a [B.sup.(1).sub.n] [+ or -]-diagram (see [5, Lemma 3.5]). Hence [PHI]([P.sub.1]), [PHI]([P.sub.2]) [member of] S([B.sup.n,2s+1]). The sequence [??] can be obtained by doubling a type [B.sup.(1).sub.n] sequence using ([m.sub.1], [m.sub.2], ..., [m.sub.n]) = (2, ..., 2,1), so by Definition 3.2 [b.sub.1] and [b.sub.2] are in the image of the embedding S that realizes [B.sup.n,2s+1]. The claim that [epsilon]([b.sub.1]) = [epsilon]([b.sub.2]) = 2s[[LAMBDA].sub.1] + [A.sub.n] can be checked explicitly.

Corollary 4.3 The KR crystal [B.sup.n,2s+1] of type [B.sup.(1).sub.n] is not perfect.

Proof: This follows directly from Proposition 4.2 using the embedding S of Definition 3.2.

Proposition 4.4 There exists a bijection, induced by [epsilon], from [B.sup.n,2s.sub.min] to [P.sup.+.sub.s]. Hence [B.sup.n,2s] is perfect of level s.

Proof: Let S be the embedding from Definition 3.2. Then we have an induced embedding of dominant weights [LAMBDA] of [B.sup.(1).sub.n] into dominant weights of [A.sup.(2).sub.2n-1] via the map S, that sends [A.sub.i] [right arrow] - [m.sub.i][[LAMBDA].sub.i].

In [22, Section 6] (see Section 4.2) the minimal elements for [A.sup.(2).sub.2n-1] were constructed by giving a [+ or -]-diagram and a sequence from the {2, ..., n}-highest weight to the minimal element. Since ([m.sub.0], ..., [m.sub.n]) = (2, ..., 2,1) and columns of height n for type [A.sup.(2).sub.2n-1] are doubled, it is clear from the construction that the [+ or -]-diagrams corresponding to weights S([LAMBDA]) are in the image of S of [+ or -]-diagrams for [B.sup.(1).sub.n] (see [5, Lemma 3.5]). Also, since under S all weights [[LAMBDA].sub.i] for 1 [less than or equal to] i < n are doubled, it follows that the sequences are "doubled" using the mi. Hence a minimal element of [B.sup.n,2s] of level s is in one-to-one correspondence with those minimal elements in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that can be obtained from doubling a [+ or -]-diagram of [B.sup.n,2s]. This implies that [epsilon] defines a bijection between [B.sup.n,2s.sub.min] and [P.sup.+.sub.s].

The automorphism [tau] of the perfect KR crystal [B.sup.n,2s] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

4.5 Type [C.sup.(1).sub.n]

In this section we consider [B.sup.r,s] of type [C.sup.(1).sub.n] for r < n.

Proposition 4.5 Let r < n. Then

min{lev([epsilon](b)) | b [member of] [B.sup.r,2s+1]} [greater than or equal to] s + 1, min{lev([epsilon](b)) | b [member of] [B.sup.r,2s]} [greater than or equal to] s.

Proof: By Definition 3.3, the crystal [B.sup.r,s] is realized inside [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The proof is similar to the proof of

Proposition 4.1 for type [B.sup.(1).sub.n].

Hence to show that [B.sup.r,2s+1] is not perfect, it is suffices to give two elements [b.sub.1], [b.sub.2] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that are fixed points under a with [epsilon]([b.sub.1]) = [epsilon]([b.sub.2]) = [LAMBDA], where lev([LAMBDA]) = 2s + 2.

Proposition 4.6 Let [b.sub.1], [b.sub.2] [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [b.sub.1] consists of s columns of the form read from bottom to top (1, 2, ..., r), s columns of the form ([bar.r], [bar.r - 1], ..., [bar.1]), and a column ([bar.r + 1], ..., [bar.2]). In [b.sub.2] the last column is replaced by (r + 2, ..., 2r + 2) if 2r + 2 [less than or equal to] n and (r + 2, ..., n, [bar.n], ..., [bar.k]) of height n otherwise. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which is of level 2s + 2.

Proof: The claim is easy to check explicitly.

Corollary 4.7 The KR crystal [B.sup.n,2s+1] of type [C.sup.(1).sub.n] is not perfect.

Proof: The {2, ..., n}-highest weight elements in the same component as [b.sub.1] and [b.sub.2] of Proposition 4.6 correspond to [+ or -]-diagrams that are invariant under [sigma]. Hence, by Definition 3.3, [b.sub.1] and [b.sub.2] are fixed points under [sigma]. Combining this result with Proposition 4.5 proves that [B.sup.r,2s+1] is not perfect.

Proposition 4.8 There exists a bijection, induced by [epsilon], from [B.sup.r,2s.sub.min] to [P.sup.+.sub.s]. Hence [B.sup.r'2s] is perfect of level s.

Proof: By Definition 3.3, [B.sup.r,s] of type [C.sup.(1).sub.n] is realized inside [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as the fixed points under [sigma]. Under the embedding S, it is clear that a dominant weight [LAMBDA] = [l.sub.0][[LAMBDA].sub.0] + [l.sub.1] [[LAMBDA].sub.1] + ... + [l.sub.n+1][[LAMBDA].sub.n+1] of type [A.sup.(2).sub.2n+1] is in the image if and only if [l.sub.0] = [l.sub.1]. Hence it is clear from the construction of the minimal elements for [A.sup.(2).sub.2n+1] as described in Section 4.2 that the minimal elements corresponding to A with [l.sub.0] = [l.sub.1] are invariant under [sigma]. By [22, Theorem 6.1] there is a bijection between all dominant weights [LAMBDA] of type [A.sup.(2).sub.2n+1] with [l.sub.0] = [l.sub.1] and lev([LAMBDA]) = 2s and minimal elements in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that are invariant under [sigma]. Hence using S, there is a bijection between dominant weights in [P.sup.+.sub.s] of type [C.sup.(1).sub.n] and [B.sup.r,2s.sub.min].

The automorphism [tau] of the perfect KR crystal [B.sup.r,2s is given by the identity.

4.6 Type [C.sup.(1).sub.n] for r = n

This case is treated in [14]. For the minimal elements, we follow the construction in Section 4.2. To every fundamental weight [[LAMBDA].sub.k] we associate a column tableau T([[LAMBDA].sub.k]) of height n whose entries are k + 1, k + 2, ..., n, [bar.n], ..., [bar.n - k + 1] (1, 2, ..., n for k = 0) reading from bottom to top. Let f([[LAMBDA].sub.k]) be defined such that T([[LAMBDA].sub.k]) = f([[LAMBDA].sub.k])[b.sub.1], where [b.sub.k] is the highest weight tableau in B(k[[LAMBDA].sub.n]). Then the minimal element b in [B.sup.n,s] such that [epsilon](b) = [LAMBDA] = [[summation].sup.n.sub.i=0] [l.sub.i][[LAMBDA].sub.i] [member of] [P.sup.+.sub.s] is constructed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The automorphism [tau] is given by

[tau]([n.summation over (i=0)] [l.sub.i][[LAMBDA].sub.i]) = [n.summation over (i=0)] [l.sub.i][[LAMBDA].sub.n-i].

4.7 Type [A.sup.(2).sub.2n]

For type [A.sup.(2).sub.2n] one may use the result of Naito and Sagaki [18, Theorem 2.4.1] which states that under their [18, Assumption 2.3.1] (which requires that [B.sup.r,s] for [A.sup.(1).sub.2n) is perfect) all [B.sup.r,s] for [A.sup.(2).sub.2n] are perfect. Here we provide a description of the minimal elements via the emebdding S into [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proposition 4.9 The minimal elements of [B.sup.r,s] of level s are precisely those that corresponding to doubled [+ or -]-diagrams in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof: In Proposition 4.8 a description of the minimal elements of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is given. We have the realization of [B.sup.r,s] via the map S from Definition 3.4. In the same way as in the proof of Proposition 4.4 one can show, that the minimal elements of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that correspond to doubled dominant weights are precisely those in the realization of [B.sup.r,s], hence e defines a bijection between [B.sup.r,s.sub.min] and [P.sup.+.sub.s].

The automorphism [tau] is given by the identity.

4.8 Type [D.sup.(2).sub.n+1] for r < n

Proposition 4.10 Let r < n. There exists a bijection [B.sup.r,s.sub.min] to [P.sup.+.sub.s], defined by [epsilon]. Hence [B.sup.r,s] is perfect.

Proof: This proof is analogous to the proof of Proposition 4.9.

The automorphism [tau] is given by the identity.

4.9 Type [D.sup.(2).sub.n+1] for r = n

This case is already treated in [14], which we summarize below. As a [B.sub.n]-crystal it is isomorphic to B(s[[LAMBDA].sub.n]). There is a description of its elements in terms of semistandard tableaux of n x s rectangular shape with letters from the alphabet A = {1 < 2 < ... < n < [bar.n] < ... < [bar.1]}. Moreover, each column does not contain both k and [bar.k]. Let [c.sub.i] be the ith column, then the action of [e.sub.i]; [f.sub.i] (i = 1, ..., n) is calculated through that of [c.sub.s] [cross product] ... [cross product] [c.sub.1] of [B([[LAMBDA]).sub.n]).sup.[cross]s]. With this realization the minimal element [b.sub.[LAMBDA]] such that [epsilon]([b.sub.[LAMBDA]]) = [LAMBDA] = [[summation].sup.n.sub.i=0] [l.sub.i][[lambda].SUB.I] [member of] [P.sup.+.sub.s] is given as follows. Let [x.sub.ij] (1 [less than or equal to] i [less than or equal to] n, j [member of] A) be the number of j in the ith row. Note that [x.sub.ij] = 0 unless i [less than or equal to] j [less than or equal to] n - i + 1. The table ([x.sub.ij]) of [b.sub.[LAMBDA]] is then given by [x.sub.ii] = [l.sub.0] + ... + [l.sub.n-i] (1 [less than or equal to] i [less than or equal to] n), [x.sub.ij] = [l.sub.j-i] (i + 1 [less than or equal to] j [less than or equal to] n), [x.sub.[bar.ij]] = [l.sub.j] + ... + [l.sub.n] (n - i + 1 [less than or equal to] j [less than or equal to] n). The automorphism [tau] is given by

[tau]([n.summation over (i=0)] [l.sub.i][[LAMBDA].sub.i]) = [n.summation over (i=0)] [l.sub.i][[LAMBDA].sub.n-i].

References

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Ghislain Fourier (1) ([dagger]), Masato Okado (2) ([double dagger]) and Anne Schilling (3) ([section])

(1) Mathematisches Institut der Universitat zu Koln, Weyertal 86-90, 50931 Koln, Germany

(2) Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

(3) Department of Mathematics, University of California, One Shields Avenue, Davis, CA 95616-8633, U.S.A.

([dagger]) Supported in part by DARPA and AFOSR through the grant FA9550-07-1-0543 and by the DFG-Projekt "Kombinatorische Beschreibung von Macdonald und Kostka-Foulkes Polynomen".

([double dagger]) Supported by grant JSPS 20540016.

([section]) Supported in part by the NSF grants DMS-0501101, DMS-0652641, and DMS-0652652.
Tab. 1: List of [c.sub.r]

                                        ([c.sub.1], ..., [c.sub.n])

[A.sub.n.sup.(1)], [D.sub.n.sup.(1)],           (1, ..., 1)
[A.sub.2n-1.sup.(2)], [A.sub.2n.sup.
(2)], [D.sub.n+1.sup.(2)]
[B.sub.n.sup.(1)]                             (1, ..., 1, 2)
[C.sub.n.sup.(1)]                             (2, ..., 2, 1)
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