Lusztig data of Kashiwara-Nakashima tableaux in type D
aa r X i v : . [ m a t h . QA ] J un LUSZTIG DATA OF KASHIWARA-NAKASHIMA TABLEAUX INTYPE D
IL-SEUNG JANG AND JAE-HOON KWON
Abstract.
We describe the embedding from the crystal of Kashiwara-Nakashimatableaux in type D of an arbitrary shape into that of i -Lusztig data associatedto a family of reduced expressions i which are compatible with the maximal Levisubalgebra of type A . The embedding is described explicitly in terms of well-known combinatorics of type A including the Sch¨utzenberger’s jeu de taquin andan analog of RSK algorithm. Introduction A Kashiwara-Nakashima tableau (KN tableau for short) is a combinatorial modelfor the crystal of a finite-dimensional irreducible representation of the classical Liealgebra, which is equal to a Young tableau in case of type A [9]. Let i be a reducedexpression of the longest element w in the Weyl group of a semisimple Lie algebra.An i -Lusztig datum is a parametrization of a PBW basis of the negative part of theassociated quantized enveloping algebra, and hence parametrizes its crystal [14, 15].Let us denote by KN λ the crystal of KN tableaux of shape λ , and denote by B i theset of i -Lusztig data.Suppose that g is a classical Lie algebra. Let l be a proper maximal Levi subal-gebra of type A , and let u be the nilradical of the parabolic subalgebra p “ l ` b ,where b is a Borel subalgebra of g . We take a reduced expression i of w such thatthe positive roots of u precede those of l with respect to the corresponding convexordering on the set of positive roots of g . Then an explicit combinatorial descriptionof the crystal embedding(1.1) KN λ (cid:31) (cid:127) / / B i , (up to a shift of weight) is given in [12] when g is of type A and in [13] when g is oftype B and C . The embedding (1.1) has several interesting features. For example, itcan be described only in terms of the well-known combinatorics of type A including Mathematics Subject Classification.
Key words and phrases. quantum groups, parabolic Verma module, crystal graphs, Kashiwara-Nakashima tableaux, Lusztig data.This work was supported by the National Research Foundation of Korea(NRF) grant funded bythe Korea government(MSIT) (No. 2019R1A2C1084833). the Sch¨utzenberger’s jeu de taquin sliding and RSK algorithm. The crystal structure B i also has a very simple description in this case.In this paper, we provide a combinatorial description of (1.1) when g is of type D (Theorem 5.8). As in [12, 13], the map is given by a composition of two embeddings,where one is the embedding of KN λ into the crystal of a parabolic Verma modulewith respect to l , say V λ , and the other is the embedding of V λ into B i (up to shiftof weights). The embedding of KN λ into V λ is given by a non-trivial sequence ofthe Sch¨utzenberger’s jeu de taquin sliding called separation , which is the main resultin this paper (Theorem 4.12). Here KN λ is replaced without difficulty by anothercombinatorial realization T λ of the crystal of integrable highest weight module,called spinor model [11]. The embedding of V λ into B i is obtained by applyingan analogue of RSK due to Burge, which is a morphism of crystals [5], and theembedding (1.1) in case of type A [12].We remark that the algorithm of separation is a generalization of the one in-troduced in [6], which yields a new combinatorial formula for the branching ruleassociated to p GL n , O n q , and the embedding of V λ into B i also has an applicationto a combinatorial description of Kirillov-Reshetikhin crystals of type D p q n [5].The paper is organized as follows. In Section 2, we introduce some necessarynotations. In Section 3, we recall the notions of KN λ and T λ , and describe theisomorphism between them. Then we describe the embedding of T λ into V λ inSection 4, and embedding of V λ into B i in Section 5 (up to shift of weights).2. Notations
Semistandard tableaux.
Let Z ` denote the set of non-negative integers. Let P be the set of partitions or Young diagrams. We let P n “ t λ P P | ℓ p λ q ď n u for n ě
1, where ℓ p λ q is the length of λ . Let P p , q “ t λ P P | λ “ p λ i q i ě , λ i P Z ` u ,where λ is the conjugate of λ . Let P p , q n “ P p , q X P n . Let λ π be the skew Youngdiagram obtained by 180 ˝ -rotation of λ .Let N be the set of positive integers with the usual linear ordering and let N bethe set consisting of i p i P N q with the linear ordering i ą j for i ă j P N . For n P N ,we put r n s “ t , . . . , n u and r n s “ t , . . . , n u , and assume that r n s Y r n s has anordering given by1 ă ă ¨ ¨ ¨ ă n ´ ă nn ă n ´ ă ¨ ¨ ¨ ă ă . For a skew Young diagram λ { µ , we denote by SST A p λ { µ q the set of semistandardtableaux of shape λ { µ with entries in a subset A of N Y N . We put SST p λ { µ q “ SST N p λ { µ q for short. For T P SST A p λ { µ q , let w p T q be the word given by readingthe entries of T column by column from right to left and from top to bottom in each USZTIG DATA OF KN TABLEAUX IN TYPE D 3 column, and let sh p T q denote the shape of T . For a P r n s and U P SST p λ π q with λ P P n , let V Ð a be the tableau obtained by applying the Schensted’s columninsertion of a into V in a reverse way starting from the rightmost column (cf. [2])2.2. Tableaux with two columns.
We also use the following notations for de-scription of our main result (cf. [6]).For a, b, c P Z ` , let λ p a, b, c q be a skew Young diagram with at most two columnsgiven by p b ` c , a q{p b q . Let T be a tableau of shape λ p a, b, c q . We denote the leftand right columns of T by T L and T R , respectively.Let T be a tableau. If necessary, we assume that it is placed on the plane with ahorizontal line L , say P L , such that any box in T is either below or above L , and atleast one edge of a box in T meets L . We denote by T body and T tail the subtableauxof T above and below L , respectively. For example, T “
121 3 ............ ............ L T body “
121 32 4 T tail “ where the dotted line denotes L .For a tableau U with the shape of a single column, let ht p U q denote the height of U and we put U p i q (resp. U r i s ) to be i -th entry of U from bottom (resp. top). Wealso write U “ p U p ℓ q , . . . , U p qq “ p U r s , . . . , U r ℓ sq , where ℓ “ ht p U q . Suppose that U is a tableau in P L . To emphasize gluing andcutting tableaux with respect to L , we also write U body ‘ U tail “ U, U a U tail “ U body . For a sequence of tableaux U , U , . . . , U m in P L , whose shapes are single columns,let us say that p U , U , . . . , U m q is semistandard along L if they form a semistandardtableau T of a skew shape with U i the i -th column of T from the left.3. Kashiwara-Nakashima tableaux and spinor tableaux
In this section, we recall two combinatorial models for the crystals of type D n ,say KN tableaux model and spinor model, and then give an explicit combinatorialdescription of the isomorphism between them. IL-SEUNG JANG AND JAE-HOON KWON
Lie algebra of type D . We assume that g is the simple Lie algebra of type D n p n ě q . The weight lattice is P “ À ni “ Z ǫ i with a symmetric bilinear form p , q such that p ǫ i | ǫ j q “ δ ij for i, j . Put I “ t , . . . , n u . The set of simple roots is t α i | i P I u , where α i “ ǫ i ´ ǫ i ` for 1 ď i ď n ´
1, and α n “ ǫ n ´ ` ǫ n , (cid:13) (cid:13) (cid:13) (cid:13)(cid:13) ✟✟✟❍❍❍ ¨ ¨ ¨ α α α n ´ α n ´ α n and the set of positive roots is Φ ` “ t ǫ i ˘ ǫ j | ď i ă j ď n u . The fundamentalweights Λ i ( i P I ) are given by Λ i “ ř ik “ ǫ k for i “ , . . . , n ´
2, Λ n ´ “ p ǫ ` ¨ ¨ ¨ ` ǫ n ´ ´ ǫ n q{ n “ p ǫ ` ¨ ¨ ¨ ` ǫ n ´ ` ǫ n q{ P n “ ! p λ , . . . , λ n q ˇˇˇ λ i P Z , λ i ´ λ i ` P Z ` , λ n ´ ě | λ n | ) . For λ P P n , we put ω λ “ n ÿ i “ λ i ǫ i . Then P ` “ t ω λ | λ P P n u is the set of dominant integral weights. We put sp ` “ `` ˘ n ˘ and sp ´ “ pp q n ´ , ´ qq for simplicity. We also identify λ P P n with a(generalized) Young diagram, which may have a half-width box on the leftmostcolumn [9, Section 6.7],Put J “ I zt n u . Let l be the Levi subalgebra of g associated to t α i | i P J u , whichis of type A n ´ . Let W be the Weyl group generated by the simple reflections s i ( i P I ) with the longest element w , and let R p w q be the set of reduced expressions of w . Recall that W acts faithfully on P by s i p ǫ i q “ ǫ i ` , s i p ǫ k q “ ǫ k for 1 ď i ď n ´ k ‰ i, i `
1, and s n p ǫ n ´ q “ ´ ǫ n and s n p ǫ k q “ ǫ k for k ‰ n ´ , n .Let U q p g q be the quantized universal enveloping algebra associated to g . ForΛ P P ` , we denote by B p Λ q the crystal associated to an irreducible highest weight U q p g q -module with highest weight Λ. For λ P P , let T λ “ t t λ u be the crystal, wherewt p t q “ λ , r e i t λ “ r f i t λ “ , and ε i p t λ q “ ϕ i p t λ q “ ´8 for i P I . We refer the readerto [7, 8] for more details of crystals.3.2. Kashiwara-Nakashima tableaux.
Suppose that λ “ p λ , . . . , λ n q P P n isgiven. The notion of Kashiwara-Nakashima tableaux (KN tableaux, for short) oftype D [9] is a combinatorial model of B p ω λ q . In this paper, we need an analogue,which is obtained from the one in [9] by applying 180 ˝ rotation and replacing i and i (resp. i with i ). For the reader’s convenience, let us give its definition and crystalstructure. USZTIG DATA OF KN TABLEAUX IN TYPE D 5
Definition 3.1.
For λ “ p λ , . . . , λ n q P P n , let T be a tableau of shape λ π withentries in r n s Y r n s such that(1) T p i, j q ğ T p i ` , j q and T p i, j q ď T p i, j ` q for each i and j ,(2) n and n can appear successively in T other than half-width boxes,(3) i and i do not appear simultaneously in the half-width boxes,where T p i, j q denotes the entry in T located in the i -th row from the bottom and the j -th column from the right. Then T is called a KN tableau of type D n if it satisfiesthe following conditions:( d -1) If T p p, j q “ i and T p q, j q “ i for some i P r n s with p ă q , then p q ´ p q` i ą λ j .( d -2) Suppose λ n ě λ j “ n . If T p k, j q “ n (resp. n ), then k is odd (resp.even).( d -3) Suppose λ n ă λ j “ n . If T p k, j q “ n (resp. n ), then k is even (resp.odd).( d -4) If either T p p, j q “ a , T p q, j q “ b , T p r, j q “ b and T p s, j ` q “ a or T p p, j q “ a , T p q, j ` q “ b , T p r, j ` q “ b and T p s, j ` q “ a with p ď q ă r ď s and a ď b ă n , then p q ´ p q ` p s ´ r q ă b ´ a .( d -5) Suppose T p p, j q “ a , T p s, j ` q “ a with p ă s . If there exists p ď q ă s such that either T p q, j q , T p q ` , j q P t n, n u with T p q, j q ‰ T p q ` , j q or T p q, j ` q , T p q ` , j ` q P t n, n u with T p q, j ` q ‰ T p q ` , j ` q , then s ´ p ď n ´ a .( d -6) It is not possible that T p p, j q P t n, n u and T p s, j ` q P t n, n u with p ă s .( d -7) Suppose T p p, j q “ a , T p s, j ` q “ a with p ă s . If T p q, j ` q P t n, n u , T p r, j q P t n, n u and s ´ q ` p ď q ă r ď s and a ă n , then s ´ p ă n ´ a .We denote by KN λ the set of KN tableaux of shape λ π .Recall that KN p q has the following crystal structure isomorphic to that of B p Λ q ¨ ¨ ¨ n ´ nn n ´ ¨ ¨ ¨ n ´ n ´ n nn ´ n ´ where a i ÝÑ b means r f i a “ b with r f i the Kashiwara operator for i P I , andwt p i q “ ǫ i , wt p i q “ ´ ǫ i . for i “ , . . . , n . On the other hand, KN sp ` and KN sp ´ have crystal structures isomorphic to those of B p Λ n q and B p Λ n ´ q which are thecrystals of spin representations with highest weights Λ n and Λ n ´ , respectively. For IL-SEUNG JANG AND JAE-HOON KWON i P I , r f i on KN sp ˘ is given by ... i ... i ` ... r f i p i ‰ n q ÝÝÝÝÝÑ ... i ` ... i ... ... n ´ n ... r f n ÝÝÝÝÑ ... nn ´ ... . (3.1)Let λ P P n be given. Let us identify T P KN λ with its word w p T q so that we mayregard KN λ Ă KN p q ˘ b N , if λ n P Z , KN sp ˘ b ` KN p q ˘ b N , if λ n R Z , where N is the number of letters in w p T q except for the one in half-width boxes.Then KN λ is invariant under r e i and r f i for i P I , and KN λ – B p ω λ q , ([9, Theorem 6.7.1]).3.3. Spinor model.
Let us briefly recall another combinatorial model for B p ω λ q (see [6, 11] for more details and examples). We keep the notations used in [6].For T P SST r n s p λ p a, b, c qq and 0 ď k ď min t a, b u , we slide down T R by k positionsto have a tableau T of shape λ p a ´ k, b ´ k, c ` k q . We define r T to to be the maximal k such that T is semistandard.For T P SST r n s p λ p a, b, c qq with r T “
0, we define E T and F T as follows:(1) E T is tableau in SST r n s p λ p a ´ , b ` , c qq obtained from T by applyingSch¨utenberger’s jeu de taquin sliding to the position below the bottom of T R , when a ą F T is tableau in SST r n s p λ p a ` , b ´ , c qq obtained from T by applying jeude taquin sliding to the position above the top of T L , when b ą E T “ and F T “ when a “ b “
0, respectively, where is a formal symbol. In general, if r T “ k , then we define E T “ E T and F T “ F T ,where T is obtained from T by sliding down T R by k positions and hence r T “ T p a q “ T | T P SST r n s p λ p a, b, c qq , b, c P Z ` , r T ď ( p ď a ď n ´ q , T p q “ ğ b,c P Z ` SST r n s p λ p , b, c ` qq , T sp “ ğ a P Z ` SST r n s pp a qq , T sp ` “ t T | T P T sp , r T “ u , T sp ´ “ t T | T P T sp , r T “ u , (3.2) USZTIG DATA OF KN TABLEAUX IN TYPE D 7 where r T of T P T sp is defined to be the residue of ht p T q modulo 2. For T P T p a q ,we define p T L ˚ , T R ˚ q when r T “
1, and p L T, R T q by p T L ˚ , T R ˚ q “ pp F T q L , p F T q R q , p L T, R T q “ pp E a ˚ T q L , p E a ˚ T q R q p a ˚ “ a ´ r T q Definition 3.2.
Let a, a be given with 0 ď a ď a ď n ´
1. We say a pair p T, S q is admissible , and write T ă S if it is one of the following cases:(1) p T, S q P T p a q ˆ T p a q or T p a q ˆ T sp with p i q ht p T R q ď ht p S L q ´ a ` r T r S , p ii q T R p i q ď L S p i q , if r T r S “ ,T R ˚ p i q ď L S p i q , if r T r S “ , p iii q R T p i ` a ´ a q ď S L p i q , if r T r S “ , R T p i ` a ´ a ` ε q ď S L ˚ p i q , if r T r S “ , for i ě
1. Here ε “ S P T sp ´ and 0 otherwise, and we assume that a “ r S , S “ S L “ L S “ S L ˚ when S P T sp .(2) p T, S q P T p a q ˆ T p q with T ă S L in the sense of (1), regarding S L P T sp ´ .(3) p T, S q P T p q ˆ T p q or T p q ˆ T sp ´ with p T R , S L q P T p q . Remark 3.3. (1) For T P T p a q , we assume that T P P L such that the subtableauof single column with height a is below L and hence equal to T tail . T “
121 3 ............ ............ L P T p q (2) Let S P T sp with ε “ r S . We may assume that S “ U L for some U P T p ε q ,where U R p i q ( i ě
1) are sufficiently large so that S “ U L “ L U . Then we mayunderstand the condition Definition 3.2(1) for p T, S q P T p a q ˆ T sp as induced fromthe one for p T, U q P T p a q ˆ T p ε q .Let B be one of T p a q p ď a ď n ´ q , T sp , and T p q . The g -crystal structure on B [11] is given as follows. Let T P B given. For i P J , we define r e i , r f i by regarding B as an l -subcrystal of Ů λ P P n SST r n s p λ q [9], where we consider the set r n s as thedual crystal of r n s that is the crystal of vector representation of l . For i “ n and T P B , we define r e n T and r f n T as follows: IL-SEUNG JANG AND JAE-HOON KWON (1) if B “ T sp , then r e n T is the tableau obtained by removing a domino nn ´ from T if it is possible, and otherwise, and r f n T is given in a similar wayby adding nn ´ ,(2) if B “ T p a q or T p q , then r e n T “ r e n ` T R b T L ˘ and r f n T “ r f n ` T R b T L ˘ regarding B Ă p T sp q b .The weight of T P B is given bywt p T q “ $&% ω n ` ř i ě m i ǫ i , if T P T p a q or T p q ,ω n ` ř i ě m i ǫ i , if T P T sp . , where m i is the number of occurrences of i in T . Then B is a regular g -crystal withrespect to r e i and r f i for i P I , and T p a q – B p ω n ´ a q p ď a ď n ´ q , T p q – B p ω n q , T p q – B p ω n ´ q , T p q – B p ω n ´ ` ω n q , T sp ´ – B p ω n ´ q , T sp ` – B p ω n q . ([11, Proposition 4.2]). Note that the highest weight element H of B is of thefollowing form:(3.3) H “ $’’’&’’’% H ‘ H p a q , if B “ T p a q with 2 ď a ď n ´ H ‘ n , if B “ T p q , H , if B “ T p q or B “ T sp ` , n , if B “ T sp ´ ,where H is the empty tableau and H p q P SST r n s pp a qq p ď a ď n ´ q such that H p a q r k s “ n ´ k ` ď k ď a ), that is, H p a q “ nn ´ n ´ a ` . (3.4)Note that the empty tableau H is an element of SST r n s pp qq .Let λ “ p λ , . . . , λ n q P P n be given. We have(3.5) ω λ “ ℓi “ ω n ´ a i ` pω n ` q p ω n q ` rω n , if λ n ě , ř ℓi “ ω n ´ a i ` pω n ` q p ω n ´ q ` rω n ´ , if λ n ă , where a ℓ ě ¨ ¨ ¨ ě a ě p is the number of i such that a i “ q , r ) (resp. ( q , r )) is given by 2 λ n “ q ` r with r P t , u (resp. ´ λ n “ q ` r with r P t , u ).Let(3.6) p T λ “ T p a ℓ q ˆ ¨ ¨ ¨ ˆ T p a q ˆ T p q ˆ q ˆ p T sp ` q r , if λ n ě , T p a ℓ q ˆ ¨ ¨ ¨ ˆ T p a q ˆ T p q ˆ q ˆ p T sp ´ q r , if λ n ă , USZTIG DATA OF KN TABLEAUX IN TYPE D 9 and regard it as a crystal by identifying T “ p . . . , T , T q P p T λ with T b T b . . . .Define T λ “ t T “ p . . . , T , T q P p T λ | T i ` ă T i for all i u , where ă is given in Definition 3.2. Then T λ Ă p T λ is invariant under r e i and r f i for i P I , and(3.7) T λ – B p ω λ q , ([11, Theorem 4.3–4.4]). The highest weight element H λ of T λ is of the form:(3.8) H λ “ H ℓ b ¨ ¨ ¨ b H b H b q b H b r ` , if λ n ě ,H ℓ b ¨ ¨ ¨ b H b H b q b H b r ´ , if λ n ă , where H i and H ˘ are the highest weight element of T p a i q and T sp ˘ given in (3.3),respectively. We call T λ the spinor model for B p ω λ q since T λ is a subcrystal of p T sp q b N for some N ě Example 3.4.
Let n “ λ “ p , , , , , q P P be given. By (3.5), we have ω λ “ ω ´ ` ω ´ “ ω ` ω , with ℓ “ p a , a , a , a q “ p , , , q . Let T “ p T , T , T , T q given by
657 6 6 5 3 ............ ...... ...... ...... ............ L T T T T where the dotted line is the common horizontal line L . Then T ă T ă T ă T ,and hence T P T λ .We also need the following in Section 4.1. Definition 3.5.
Let B be one of T p b q p ď b ă n q , T sp , and T p q . For p T, S q P T p a q ˆ B with a P Z ` , we write T Ÿ S if the pair p R T, S L q forms a semistandardtableau of a skew shape, where we assume that R T and S L are arranged along L asfollows: R T “ p . . . , R T p a ` qq ‘ p R T p a q , . . . , R T p qq ,S L “ p . . . , S L p b ` qq ‘ p S L p b q , . . . , S L p qq . Here we understand S in the sense of Remark 3.3 and put b “ ht p S tail q when S P T sp ´ or T p q . Isomorphisms.
Let us give an explicit description of the isomorphisms be-tween KN λ and T λ for λ P P n (cf. [13, Section 3.3] for type B n and C n ).Let B be one of T p a q p ď a ď n ´ q , T sp ˘ , and T p q . For T P B , we define atableau r T as follows: Case 1 . Suppose that B “ T sp ˘ . Let r T be the unique tableau in SST r n s pp n qq suchthat i appears in T if and only if i appears in r T for 1 ď i ď n . Case 2 . Suppose that B “ T p a q p ď a ă n ´ q or T p q .(i) First, let Ă R T be the unique tableau in SST r n s p m q with m “ n ´ ht p R T q such that i appears in Ă R T if and only if i does not appear in R T for i P r n s .(ii) Then define r T to be the tableau of single column obtained by puttingthe single-column tableau consisting of nn with height b ´ r T between L T and Ă R T , where L T is located below Ă R T . Example 3.6.
Let n “ T P T p q be T in Example 3.4 with sh p T q “ λ p , , q and r T “
1, where we have T L “ , T R “ , L T “ , R T “ . Then Ă R T is given by R T “ ÝÝÝÝÝÑ “ Ă R T . and hence T “
655 34 232
ÝÝÝÝÝÑ “ r T .
Note that since b “ r T “
1, there is no domino nn in r T . On the other hand,if T P T sp ´ is given as follows, then we have T “ ÝÝÝÝÝÑ “ r T .
USZTIG DATA OF KN TABLEAUX IN TYPE D 11
Lemma 3.7.
The map sending T to r T gives an isomorphisms of crystals Φ : B / / $&% KN sp ˘ , if B “ T sp ˘ , KN p n ´ a q , if B “ T p a q or T p q . Proof.
Case 1. B “ T sp ˘ . It is straightforward to see that Φ is a weight preservingbijection and by (3.1), it is a morphism of crystals. Hence Φ is an isomorphism. Case 2.
Suppose that B “ T p a q p ď a ă n ´ q or T p q . Let T P B given. Firstwe claim r T P KN p n ´ a q . Suppose that r T R KN p n ´ a q . Then by the condition ( d -1)there exists i P r n s such that(3.9) p q ´ p q ` i ď n ´ a p p ă q q . Put x “ n ´ a ´ q and y “ p . We note that x is the number of entries in r n s smallerthan i in r T , and y is the number of entries in r n s equal to or larger than i in r T . Take k such that L T p k q “ i . Then we have L T p k q ą R T p k q by (3.9). This contradicts tothe fact that the pair p L T, R T q forms a semistandard tableau when the two columnsare placed on the common bottom line. Hence r T P KN p n ´ a q .Second we show that T ” l r T , where ” l denotes the crystal equivalence as elementsof l -crystals. By the construction of Ă R T , it is not difficult to check that Ă R T ” l R T (more precisely as elements of sl n -crystals). Put D to be the single-column tableauconsisting of the domino nn with height b ´ r T . By the tensor product rule of crystals,we see that t D u is the crystal of the trivial representation of l . This implies that Ă R T ” l Ă R T b D and thus T ” l R T b L T ” l Ă R T b L T ” l Ă R T b D b L T ” l r T Next we claim that r T “ r f n r T , where T “ r f n T . Let T P SST r n s p λ p a, b, c qq and T P SST r n s p λ p a, b , c qq . Let us consider the case when r f n p T R b T L q ‰ and r f n p T R b T L q “ T R b p r f n T L q . The proof of the other cases is similar. In this case, wehave b “ b ´ c “ c ` r f n , and that T L r s and T R r s must satisfythat n ´ ď T L r s , T R r s ď n ´ . Then we have r f n r T “ R T b r f n p D q b L T, if b ą , r f n p Ă R T q b L T, if b “ , where r f n p D q is obtained from D by replacing nn by nn ´ at the bottom of D . Notethat the bottom entry of r f n p Ă R T q is given by n , if T R r s “ n, n ´ , if T R r s “ n ´ . On the other hand, we can check that L T is obtained from L T by putting the domino nn ´ (resp. n if T R r s “ n , and n ´ if T R r s “ n ´ ) on the top of L T when b ą b “ r f n r T is equal to r T .Consequently, Φ is a morphism of crystals. Since Φ is injective and sends thehighest weight elements of T p a q to that of KN p n ´ a q , Φ is an isomorphism. (cid:3) Next let us describe the inverse map of Φ. Let T P KN p a q Y KN sp ˘ (0 ă a ď n )be given. Then we define Ψ a p T q and Ψ sp ˘ p T q if T P KN p a q and T P KN sp ˘ respectively as follows:(1) Let T ` (resp. T ´ ) be the subtableau in T with entries in r n s (resp. r n s )except for dominos nn (2) Let Ă T ` be the single-column tableau with height n ´ ht p T ` q such that i appears in T ` if and only if i does not appear in Ă T ` .(3) We define Ψ a p T q and Ψ sp ˘ p T q byΨ sp ˘ p T q “ T ´ , Ψ a p T q “ F n ´ a ´ ǫ p T ´ , Ă T ` q , (3.10) where ǫ “ , if ht p Ă T ` q ´ a is even , , if ht p Ă T ` q ´ a is odd . We note that Ψ a p T q has residue 1 if ht p Ă T ` q ´ a is odd, otherwise 0.It is not difficult to check that the map Ψ a (resp. Ψ sp ˘ ) is the inverse of Φ. Henceby Lemma 3.7, we have the following. Lemma 3.8.
The maps Ψ a and Ψ sp ˘ are isomorphisms of crystals Ψ sp ˘ : KN sp ˘ / / T sp ˘ , Ψ a : KN p a q / / T p n ´ a q p ă a ď n q , (cid:3) Now we consider the isomorphism for any λ P P n . Let µ “ p a ℓ , . . . , a q , where a , . . . , a ℓ are given in (3.5), with ℓ “ µ . For T P KN λ , let p T ℓ , . . . , T q , if λ n P Z , p T ℓ , . . . , T , T q , if λ n R Z , denote the sequence of columns of T , where T is the column of T with half-widthboxes, and T , T , . . . are the other columns enumerated from right to left. USZTIG DATA OF KN TABLEAUX IN TYPE D 13
Theorem 3.9.
For λ P P n , the map (3.11) Ψ λ : KN λ / / T λ defined by Ψ λ p T q “ p Ψ µ ℓ p T ℓ q , . . . , Ψ µ p T qq , if λ n P Z , p Ψ µ ℓ p T ℓ q , . . . , Ψ µ p T q , Ψ sp ˘ p T qq , if λ n R Z , is an isomorphism of crystals from KN λ to T λ , where we take Ψ sp ` and Ψ sp ´ if λ n ě and λ n ă , respectively. Proof.
By Lemma 3.8, the map Ψ λ is an embedding of crystals into p T λ . Also themap Ψ λ sends the highest weight element of KN λ to the one of T λ (cf. (3.8)). Thenby (3.7), the image of Ψ λ is isomorphic to T λ . (cid:3) Example 3.10.
Let λ “ p , q P P be given. Consider T “ P KN λ . Put T “ , T “ , T “ , T “ . By definition of pp T i q ´ , Ć p T i q ` q , p p T q ´ , Č p T q ` q “ ˜ , ¸ , p p T q ´ , Č p T q ` q “ ˜ , ¸ , p p T q ´ , Č p T q ` q “ ˜ , ¸ , p p T q ´ , Č p T q ` q “ ˜ , ¸ . Since ht p Ć p T i q ` q ´ ht p T i q is odd for i “ , , ,
4, we have by (3.10) Ψ p T q “ , Ψ p T q “ , Ψ p T q “ , Ψ p T q “
655 34 232 . Hence, we obtain Ψ λ p T q “
657 6 6 5 3 ......... ...... ...... ...... ............ P T λ . Embedding into the crystal of parabolic Verma module
In this section, we describe an embedding of T λ into the crystal of a parabolicVerma module corresponding to λ P P n with respect to l . The embedding maps T P T λ to a pair of semistandard tableaux, which can be described in terms of acombinatorial algorithm called separation .4.1. Sliding.
Suppose that λ P P n is given. The algorithm of separation consists ofcertain elementary steps denoted by S j ( j “ , , . . . ) as operators on T P T λ , eachof which moves a tail in T by one position left based on the Sch¨utzeberger’s jeu detaquin.Note that T λ is a subcrystal of p T sp q b N for some N . We may identify p T sp q b N with E N : “ ğ p u N ,...,u qP Z n ` SST r n s p u N q ˆ ¨ ¨ ¨ ˆ SST r n s p u q . In order to give a precise description of the operator S j , we use an p l , sl N q -bicrystalstructure on E N as in [13, Lemma 5.1]: The l -crystal structure on E N with respectto r e i and r f i for i P I is naturally induced from that of p T sp q b N . On the other hand,the sl N -crystal structure is defined as follows. Let p U N , . . . , U q P E N given. For1 ď j ď N ´ X “ E , F , we define X j p U N , . . . , U q “ $&% p U r , . . . , X p U j ` , U j q , . . . , U q , if X p U j ` , U j q ‰ , , if X p U j ` , U j q “ . where X p U j ` , U j q is understood to be X U for some U P SST r n s p λ p a, b, c qq with r U “ U L “ U j ` and U R “ U j (see Section 3.3).Let l be the number of components in p T λ in (3.6) except T sp ˘ . Consider anembedding of sets(4.1) T λ / / E l ` T “ p T l , . . . , T , T q ✤ / / p T L l , T R l , . . . , T L , T R , T q , USZTIG DATA OF KN TABLEAUX IN TYPE D 15 where T is regarded as T P t H u , if λ n P Z , T sp ˘ , if λ n R Z . Here tHu is the crystal of trivial module. We identify T “ p T l , . . . , T , T q P T λ with its image U “ p U l , . . . , U , U q under (4.1) so that T “ U and p T i ` , T i q isgiven by(4.2) p T i ` , T i q “ p U j ` , U j ` , U j , U j ´ q “ p T L i ` , T R i ` , T L i , T R i q , with j “ i for 1 ď i ď l ´ S j on T for j “ i for 1 ď i ď l ´ S j “ $&% F a i j , if T i ` Ÿ T i , E j E j ´ F a i ´ j F j ´ , if T i ` Ž T i , where S j is understood as the identity operator when a i “
0, and Ÿ is given inDefinition 3.5. Remark 4.1.
The operator S j in (4.3) agrees with the one in [6], which is definedonly on the set of l -highest weight elements.The following lemma is crucial in Section 4.2. Lemma 4.2.
Let T “ p . . . , T i ` , T i , . . . q P T λ be given. (1) We have S j T “ p . . . , U j ` , r U j ` , r U j , U j ´ , . . . q for some r U j ` and r U j , and p U j ` , r U j ` , r U j , U j ´ q is semistandard along L . (2) Suppose that r e k T ‰ for some k P J and put S “ r e k T “ p . . . , S i ` , S i , . . . q .Then T i ` Ÿ T i if and only if S i ` Ÿ S i . Proof.
The proof is given in Section 4.4. (cid:3)
Separation when λ n ě . Let us assume λ P P n with λ n ě
0. The case when λ n ă T “ p T l , . . . , T , T q P T λ be given. Since T i P P L for 0 ď i ď l , we mayconsider the p l ` q -tuples(4.4) p T body ℓ , . . . , T body , T body q , p T tail ℓ , . . . , T tail , T tail q to form tableaux in P L . But in general, they are not necessarily semistandard along L , and p T body l , . . . , T body , T body q may not be of a partition shape along L . So insteadof cutting T with respect to L directly as in (4.4), we will introduce an algorithm toseparate T into two semistandard tableaux, which preserves l -crystal equivalence.More precisely, we introduce an algorithm to get a semistandard tableau T in P L such that (S1) T is Knuth equivalent to T , that is, T ” l T ,(S2) T tail P SST r n s p µ q and T body P SST r n s p δ π q for some δ P P p , q n , where µ P P n is given by(4.5) µ “ p a ℓ , . . . , a q with a i as in (3.5).We call this algorithm separation which can be viewed as a generalization of the onein [6] (see [13] for type B and C ). Let us explain this with an example before wedeal it in general. Example 4.3.
Let T “ p T , T , T , T q P T p , , , , , q be given in Example 3.4. ............ ...... ...... ...... ............ L U U U U U U U U where p U , . . . , U q denotes the image of p T , T , T , T q under (4.1).First we consider Ÿ in Definition 3.5 on p T i ` , T i q for 1 ď i ď
3. Then we cancheck that T Ž T , T Ž T and T Ÿ T . By (4.3), we have S “ E E F F , S “ E E F F , S “ F . Now, we apply these operators S , S , and then S to T to have ............ ...... ...... ...... ............ L U r U r U r U r U r U r U U We observe that (recall Definition 3.2)(4.6) p r U , r U q ă p r U , r U q ă p r U , r U q , (we will show in in Lemma 4.4 that this holds in general). So we can apply the aboveprocess to (4.6), and repeat it until there is no tail to move to the left horizontally. USZTIG DATA OF KN TABLEAUX IN TYPE D 17
Consequently we have
656 6 6 5 3 ............ ............ L Hence we obtain two semistandard tableaux of shape δ π and µ , where δ “p , , , q and µ “ p , , q . (cid:3) Now let T “ p T l , . . . , T , T q P T λ be given and let U “ p U l , . . . , U , U q be itsimage under (4.1). We use the induction on the number of columns in T to define T . If n ď
3, then let T is given by putting together the columns in U horizontallyalong L .Suppose that n ě
4. First, we consider S . . . S l ´ T “ S . . . S l ´ U “ p U l , r U l ´ , . . . , r U , U , U q P E l ` , and let r U “ p r U l ´ , . . . , r U , U , U q P E l . Note that applying S i to S i ` . . . S l ´ T for 1 ď i ď l ´ Lemma 4.4.
Let r λ P P n be such that ω λ ´ ω r λ “ ω n ´ a with a “ ht p U tail l q . Thenthere exits a unique r T P T r λ such that the image of r T under (4.1) is equal to r U . Proof.
There exists an l -highest weight element H P T λ such that H “ r e i . . . r e i r U for some i , . . . , i r P I . Put U “ p U l ´ , . . . , U q and let H be obtained from H byremoving its leftmost column, say U l . By tensor product rule of crystals, H is alsoan l -highest weight element. Identifying U “ U b U l We observe that H b U l “ r e i . . . r e i r U “ r e i . . . r e i r ` U b U l ˘ “ `r e j . . . r e j s U ˘ b p r e k . . . r e k t U l q where t i , . . . , i r u “ t j , . . . , j s u Y t k , . . . , k l u . Hence H “ r e j . . . r e j s U , and r U “ S . . . S l ´ U “ S . . . S l ´ ´ r f j s . . . r f j H ¯ “ r f j s . . . r f j ` S . . . S l ´ H ˘ , since E l is an p l , sl l q -bicrystal. Note that the operator S . . . S l ´ (4.3) is well-defined on H by Remark 4.1 and Lemma 4.2(2). Then S . . . S l ´ H P T r λ by [6,Lemma 3.10, 3.18], which implies that r U P T r λ . (cid:3) Put T “ r T , which is given in Lemma 4.4. By induction hypothesis, there existsa tableau T satisfying (S1) and (S2) associated to T . Then we define T to be thetableau in P L obtained by putting together the leftmost column of T , that is, U l ,and T along L . By definition, sh p T q “ η is of the following form:(4.7) η “ δ π µ L for some δ P P p , q n . Proposition 4.5.
Under the above hypothesis, T satisfies (S1) and (S2). Proof.
By definition, it is clear that T ” l T , which implies (S1). By Lemma 4.2, p U body l , T body q and p U tail l , T tail q are semistandard along L , which implies (S2). (cid:3) Separation when λ n ă . We consider the algorithm for separation when λ n ă λ n ă
0. Recall that ´ λ n “ q ` r with r P t , u . For T P T λ , we may write T “ p T l , . . . , T m ` , T m , . . . , T , T q , for some m ě T i P T p a i q for some a i p m ` ď i ď l q , T i P T p qp ď i ď m q and T P T sp ´ (resp. T “ H ) if r “ r “ T with U “ p U l , . . . , U m , U m ´ , . . . , U , U q under (4.1). Remark 4.6.
For the spin column U m , we apply the sliding algorithm in Section4.1 as follows. Let U “ H p n q in (3.4). Consider the pair p U m ` , U m ` , U m , U q , where we regard U “ U ‘ H P P L and p U m , U q P T p ε q as in Remark 3.3(2). ByLemma 4.2, we have p U m ` , U m ` , U m , U q S m ÝÝÝÝÝÑ p U m ` , r U m ` , r U m , U q , USZTIG DATA OF KN TABLEAUX IN TYPE D 19 for some r U m ` and r U m , and by our choice of U , we have p U m ` , U m ` , U m q ” l p U m ` , r U m ` , r U m q . Now, we use the induction on the number of columns in T to define T satisfying(S1) and (S2) where µ P P n in this is given by(4.8) µ “ p a ℓ , . . . , a , , . . . , loomoon ´ λ n q . If n ď
3, then let T be given by putting together the columns in U along L .Suppose that n ě
4. First, we consider S l ´ S l ´ . . . S m T , where S m is under-stood as in Remark 4.6.. Then we have(4.9) S l ´ S l ´ . . . S m T “ p U l , r U l ´ , . . . , r U m ` , r U m , U m ´ , . . . , U , U q , for some r U i for 2 m ď i ď l ´
1. Let r U “ p r U l ´ , . . . , r U m ` , r U m , U m ´ , . . . , U , U q . The following is an analogue of Lemma 4.4 for λ n ă Lemma 4.7.
Let r λ be such that ω λ ´ ω r λ “ ω n ´ a with a “ ht p U tail l q . Then thereexists a unique r T P T r λ such that the image of r T is equal to r U under (4.1) . Proof.
Let U “ p U l , U l ´ , . . . , U m ` , U m , U q . Let ν “ wt p U q and let r ν begiven by ν ´ r ν “ ω n ´ a with a “ ht p U tail l q . By Lemma 4.4 and Remark 4.6, thereexists a unique r S P T r ν whose image under (4.1) is Ă U “ p r U l ´ , . . . , r U m , U q . Bysemistandardness of p U m , U m ´ q (cf. Remark 4.6), we have p r U m ` , r U m q ă p U m ´ , U m ´ q . Note that we may regard p U j ´ , U j ´ q P T p q p ď j ď m q , and p U , U q P T p q if U ‰ H , p U , U q P T sp ´ otherwise. Therefore there exists a unique r T P T r λ whichis equal to r U under (4.1). (cid:3) Put T “ r T given in Lemma 4.7. By induction hypothesis, there exists a tableau T satisfying (S1) and (S2) associated to T . Then we define T to be the tableau in P L obtained by putting together the leftmost column of T , that is, U l , and T along L . By definition, sh p T q “ η is of the form (4.7) with µ in (4.8). Proposition 4.8.
Under the above hypothesis, T satisfies (S1) and (S2). Proof.
It follows from the same argument as in the proof of Proposition 4.5 withLemma 4.7. (cid:3)
Example 4.9.
Let n “ λ “ ` , , , , ´ ˘ . Then we have ω λ “ ω ` ω ` ω , T λ Ă T p q ˆ T p q ˆ T sp ´ . Let us consider T “ p T , T , T q P T λ given by
55 3 4 ............ ...... ...... ............ L T T T We regard T as in Remark 3.3(1). Then we have T Ž T and T Ž T . By applying(4.9) (cf. Remark 4.6) we obtain ............ ...... ...... ............ T T T S S ÝÝÝÝÝÑ
55 44 35 3 2 .............................. ............ ............ U r U r U r U r U where U “ H p q is the single-column tableau consisting of the numbers in gray.Finally, we apply S to p U , r U , r U , r U , r U q and then we have T given by T “
545 3 ............ ............ L where T body (resp. T tail ) is the semistandard tableau located above L (resp. belowL) whose shape is p , , , q π (resp. p , , , q ).4.4. Proof of Lemma 4.2.
Let B be one of T p a q p ď a ď n ´ q and T sp in (3.2).When B “ T sp , we regard an element of B as in the sense of Remark 3.3 (2). For USZTIG DATA OF KN TABLEAUX IN TYPE D 21 T P B , we define(4.10) s T : “ max ď s ď ht p T R q s | T L p s ´ r T ´ ` a q ą T R p s q ( Y t u , where T L r k s : “ ´8 for k ď
0. Note that a “ r T when B “ T sp .Let p T, S q P T p a q ˆ B be an admissible pair such that T P SST r n s p λ p a , b , c qq , S P SST r n s p λ p a , b , c qq for a i P Z ` and b i , c i P Z ` p i “ , q with a ď a . If the pair p T, S q satisfiesht p T R q ą ht p S L q ´ a , then put S L r s : “ ´8 . Note that above inequality occurs only for the case r T ¨ r S “ Lemma 4.10. (1) If r T ¨ r S “ p resp. r T ¨ r S “ q , then T Ÿ S ` resp. T Ÿ p S L ˚ , S R ˚ q ˘ . (2) If r T ¨ r S “ , then T Ÿ S is equivalent to the following condition: (4.11) T R p k q ď S L p k ´ ` a q for s T ď k ď ht p T R q . Proof. (1) : If r T ¨ r S “ r T ¨ r S “ T Ÿ S (resp. T Ÿ p S L ˚ , S R ˚ q ) followsimmediately from Definition 3.2 (iii).(2) : Assume that r T ¨ r S “
1. The relation T Ÿ S implies T R p k q “ R T p k ´ ` a q ď S L p k ´ ` a q for s T ď k ď ht p T R q . Conversely, we assume that (4.11) holds. Note that by definition of S L ˚ ,(4.12) S L ˚ p k q “ S L p k q , if there exists k such that 1 ď k ď s S ´ ` a . Now we consider two cases. Case 1 . s T ą s S . If s S ď k ă s T , then by Definition 3.2 (ii),(4.13) T R p k q “ T R ˚ p k q ď L S p k q “ S L p k ´ ` a q . Combining (4.11), (4.12), (4.13) and Definition 3.2 (iii), we have R T p k ` a ´ a q ď S L p k q for 1 ď k ď ht p S L q . Case 2 . s T ď s S . In this case, the relation T Ÿ S follows directly from (4.11),(4.12) and Definition 3.2 (iii). (cid:3) Lemma 4.11.
Assume that r T ¨ r S “ . For ď k ď ht p T R q , (1) T R p k q ď S L ˚ p k q . (2) If T Ÿ S , then T R p k q ď S L p k q . Proof.
By (4.13) and Definition 3.2 (iii), in any case, we have(4.14) T R p s S q ď S L p s S ´ ` a q ď S R p s S q “ S L ˚ p s S ´ ` a q . Then (1) follows from Definition 3.2 (ii)–(iii) and (4.14). By the same argument inthe proof of Lemma 4.10 (2), we obtain (2). (cid:3)
Under the map (4.2), put p T, S q “ p U , U , U , U q , S p T, S q “ p r U , r U , r U , r U q . (4.15)Here S is the operator given as in (4.3). For 1 ď i ď
4, we regard a tableau U i asfollow: U body “ U a H , U body “ U a S tail ,U body “ U a H , U body “ U a T tail , where T tail “ p T L p a q , . . . , T L p qq and S tail “ p S L p a q , . . . , S L p qq . Then weconsider (4.15) in P L . For simplicity, put h “ ht p T R q , g “ ht p S R q . We consider some sequences given inductively as follows:(i) Define a sequence v ă ¨ ¨ ¨ ă v h by v “ min ď k ď a k ˇˇ T L p k q ď T R p q ( ,v s “ min v s ´ ` ď k ď s ` a k ˇˇ T L p k q ď T R p s q ( . (4.16)(ii) Define a sequence w ă ¨ ¨ ¨ ă w h by w h “ max h ď k ď h ` a k ˇˇ T R p h q ď S L p k q ( ,w t “ max t ď k ď w t ` ´ k ˇˇ T R p t q ď S L p k q ( . (4.17)(iii) Define a sequence x ă ¨ ¨ ¨ ă x g by x “ min ď k ď a k ˇˇ S L p k q ď S R p q ( ,x u “ min x u ´ ` ď k ď u ` a k ˇˇ S L p k q ď S R p u q ( . (4.18) Proof of Lemma 4.2 (1).
By Lemma 4.10 (1), the proof for the case r T ¨ r S “ r T ¨ r S “ USZTIG DATA OF KN TABLEAUX IN TYPE D 23
Case 1 . T Ÿ S . In this case, S “ F a . This implies that r U “ U and r U “ U .By Lemma 4.11 (2), p T R , S L q P SST r n s p λ p , b, c qq , where b “ a ` c ´ b ´ c and c “ b ` c . Since T ă S with T Ÿ S , c ´ b ´ c ě F a p T, S q is well-defined. Now we show that p U , r U q and p r U , U q aresemistandard along L . (1) p U , r U q is semistandard along L . For 1 ď k ď ht p T R q satisfying v k ě a ´ a `
1, the relation T Ÿ S implies(4.19) v k ´ a ` a ď w k . (i) Assume that r U p k q “ T R p k q for some k . By definition of w k , we have(4.20) k “ w k . If v k ě a ´ a `
1, then U p k ` a ´ a q “ T L p w k ` a ´ a q by (4.15) and (4.20) ď T L p v k q by (4.19) ď T R p k q “ r U p k q by (4.16)(4.21) If v k ă a ´ a `
1, then(4.22) U p k ` a ´ a q “ T L p k ` a ´ a q ă T L p v k q ď T R p k q “ r U p k q . (ii) Assume that r U p k q ‰ T R p k q for any k . In this case, we have r U p k q “ S L p k q . Then U p k ` a ´ a q “ T L p k ` a ´ a qď R T p k ` a ´ a q by definition of R T ď S L p k q “ r U p k q by T Ÿ S (4.23)By (4.21), (4.22) and (4.23), p U , r U q is semistandard along L . (2) p r U , U q is semistandard along L . By (4.17), (4.18) and Definition 3.2 (ii), wehave(4.24) x k ď w k for 1 ď k ď s T ´ . Also the relation T Ÿ S implies(4.25) k ´ ` a ď w k for k ě s T . (i) For 1 ď k ď s T ´ r U p k q “ S L p w k q by (4.17) ď S L p x k q by (4.24) ď S R p k q “ U p k q by (4.18)(ii) For k ě s T , (4.25) implies r U p k q “ S L p w k q ď S L p k ´ ` a q ď S R p k q “ U p k q . Note that S L p k ´ ` a q ď S R p k q holds since r S “ p r U , U q is semistandard along L . Case 2 . T Ž S . In this case, S “ E E F a ´ F . Note that by definition F p S L , S R q “ p S L ˚ , S R ˚ q . By Lemma 4.11 (1), p T R , S L ˚ q P SST r n s p λ p , b, c qq , where b “ a ` c ´ b ´ c ` c “ b ` c . Note that by Definition 3.2 (i), b “ a ´ ` t c ´ p b ` c ´ qu ě a ´ . Therefore, F a ´ F p T, S q is well-defined. Put F a ´ F p T, S q “ p U , U , U , U q . Note that U “ U by definition of F a ´ F . We use sequences p w k q and p x k q in(4.17) and (4.18) replacing S L , S R and a with S L ˚ , S R ˚ and a ´
1, respectively. (1) p U , U q is semistandard along L . By Lemma 4.10 (1), we use the similarargument in the proof of
Case 1 (1) . (2) p U , U q is semistandard along L . We observe E a p S L ˚ , S R ˚ q “ E a F p S L , S R q “ E a ´ p S L , S R q “ p L S, R S q . Therefore we use the similar argument in the proof of
Case 1 (2) .Note that (4.14) implies that S R p s S q is contained in U . Then the operator E on p U , U , U , U q moves S R p s S q by one position to the right. Therefore we have(4.26) r U “ U . On the other hand, (4.13) and Definition 3.2 (iii) implies that the operator E on E p U , U , U , U q moves U p w k q “ T R p k q for some k ě s T . USZTIG DATA OF KN TABLEAUX IN TYPE D 25 by one position to the right. By the choice of s T and s S (4.10) with (4.26), p U , r U q and p r U , U q are semistandard along L .We complete the proof of Lemma 4.2 (1). (cid:3) Proof of Lemma 4.2 (2).
Put r e k p T, S q “ p T , S q . Then it is not difficult to checkthat(4.27) r T “ r T , r S “ r S . pñq Assume that(4.28) r e k p T, S q “ p T, r e k S q and r e k S “ p r e k S L , S R q p k P J q . Otherwise it is clear that T Ÿ S .If r T ¨ r S “
0, then Lemma 4.10 (1) and (4.27) implies that T Ÿ S holds.If r T ¨ r S “
1, then suppose T Ž S . By Lemma 4.10 (2), there exists s ě s T suchthat T R p s q “ k which contradicts to (4.28) by the tensor product rule. Hence we have T Ÿ S . pðq It follows from the similar argument of the previous proof. (cid:3)
Embedding.
For λ P P n , let V λ : “ ˜ Ů δ P P p , q n SST r n s ` δ π ˘¸ ˆ SST r n s p µ q , where µ P P n is given by (4.5) if λ n ě
0, and by (4.8) otherwise. Recall that V : “ ğ δ P P p , q n SST r n s p δ π q , has a g -crystal structure (see [10, Section 5.2] for details). On the other hand, weregard the l -crystal S µ : “ SST r n s p µ q , as a g -crystal, by defining r e n T “ r f n T “ with ϕ n p T q “ ε n p T q “ ´8 for T P SST r n s p µ q . Then we may regard V λ as a g -crystal by letting V λ “ V b S µ , which can be viewed as the crystal of a parabolic Verma module induced from ahighest weight l -module with highest weight λ . The following is the main theoremin this section. Theorem 4.12.
The map (4.29) T λ / / V λ b T rω n T ✤ / / T body b T tail b t rω n is an embedding of crystals, where r “ p λ, ω n q . Before proving Theorem 4.12, let us recall the actions of r e n and r f n on T λ and V λ in more details. We let vd “ n n ´ be the vertical domino with entries n and n ´ T “ p T l , . . . , T , T q P T λ is given, where T “ p U ℓ , . . . , U i , . . . , U , U q under (4.1). We define a sequence σ “ p σ , σ , . . . , σ l q by(4.30) σ i “ $’&’% ` if U i “ H or U i r s ě n ´ , ´ if vd is located in the top of U i , ¨ otherwise , where we put σ “ ¨ if U “ H . Let σ red be the (reduced) sequence obtainedfrom σ by replacing the pairs of neighboring signs p` , ´q (ignoring ¨ ) with p ¨ , ¨ q as far as possible. If there exists a ´ in σ red , then we define r e n T to be the oneobtained by removing vd in U i corresponding to the rightmost i such that σ i “ ´ in σ red . We define r e n T “ , otherwise. Similarly, we define r f n T by adding vd in U i corresponding to the leftmost i such that σ i “ ` in σ red .Next, suppose that T P V λ is given. We define a sequence τ “ p τ , τ , . . . q by(4.30). Note that τ is an infinite sequence where τ i “ ` for all sufficiently large i .Then we define τ red and hence r e n T and r f n T in the same way as in T λ . Lemma 4.13.
Under the above hypothesis, let σ “ p σ , . . . , σ l q be the subsequenceof τ red consisting of its first l ` components. If we ignore ¨ , then σ red “ σ red . Proof.
We first consider a pair p T i ` , T i q in T . Let p σ j ´ , σ j , σ j ` , σ j ` q be thesubsequence of σ corresponding to p T i ` , T i q with j “ i . Let S j T “ p . . . , U j ` , r U j ` , r U j , U j ´ , . . . q (see Lemma 4.2). We denote by p σ j ´ , r σ j , r σ j ` , σ j ` q the sequence defined by (4.30)corresponding to p U j ` , r U j ` , r U j , U j ´ q . Case 1. r i ` r i “
0. Note that S j “ F a i j by Lemma 4.10 (1). Then we have p σ j ` , σ j q “ p r σ j ` , r σ j q by the similar argument in the proof of [13, Theorem 5.7]. USZTIG DATA OF KN TABLEAUX IN TYPE D 27
Case 2. r i ` r i “
1. The relation between the two pairs p σ j ` , σ j q and p r σ j ` , r σ j q isgiven in Table 1. p σ j , σ j ` q p r σ j , r σ j ` qp` , `q p` , `qp` , ´q p` , ´q or p ¨ , ¨ qp` , ¨ q p` , ¨ q or p ¨ , `qp´ , `q p´ , `qp´ , ´q p´ , ´qp´ , ¨ q p´ , ¨ q or p ¨ , ´qp ¨ , `q p ¨ , `q or p` , ¨ qp ¨ , ´q p ¨ , ´q or p´ , ¨ qp ¨ , ¨ q p ¨ , ¨ q Table 1.
The relation between p σ j , σ j ` q and p r σ j , r σ j ` q when r i ` r i “ U : “ S . . . S l ´ T : “ p U l , r U l ´ , . . . , r U , U , U q and let σ be the sequencegiven by (4.30) corresponding to U . Then we have σ “ p σ l , r σ l ´ , . . . , r σ , σ , σ q . Itis straightforward to check that σ red “ σ red . Note that σ l “ σ l by definition. ByLemma 4.4 and 4.7, we may use an inductive argument for p r U ℓ ´ , . . . , r U , U , U q to have σ red “ σ red “ σ red . This completes the proof. (cid:3) Proof of Theorem 4.12 . We will use the following notations under the identification(4.1) in this proof.(1) r T “ p r U l , r U l ´ , . . . q : the sequence of tableaux with a column shape ob-tained from T by shifting the tails one position to the left as in Sections 4.2and 4.3.(2) T “ p r U l ´ , r U l ´ , . . . q : the sequence of tableaux obtained from r T by re-moving the column r U l .(3) T “ p U l , U l ´ , . . . q : the sequence obtained from T by applying r f n , that is, T “ r f n T . (4) r T : the sequence obtained from T by shifting the tails one position to the leftas in Sections 4.2 and 4.3 and then removing the left-most column of T .(5) r σ “ p r σ , r σ , . . . , r σ l q : the sequence of (4.30) associated to r T .Recall that T P T r λ by Lemmas 4.4 and 4.7, where r λ satisfies ω λ ´ ω r λ “ ω n ´ a with a “ ht p U tail l q .Let us denote by χ λ the map in (4.29). First we show that χ λ is injective.Suppose that χ λ p T q “ χ λ p T q for some T , T P T λ . There exists i , . . . , i r P J such that r e i . . . r e i r χ λ p T q “ r e i . . . r e i r χ λ p T q is an l -highest weight element. Note that χ λ is a morphism of l -crystals by Proposition 4.5 (cf. Section 4.5). Therefore, we have χ λ p r e i . . . r e i r T q “ χ λ p r e i . . . r e i r T q . By [6, Proposition 3.4, Lemma 6.5], we have r e i . . . r e i r T “ r e i . . . r e i r T . Then T “ T . Hence χ λ is injective.Now it remains to show that(4.31) r f n T ‰ and χ λ p T q ‰ for T P T λ ùñ χ λ p r f n T q “ r f n χ λ p T q . If r f n acts on U l or U k , where k “ U ‰ H , and k “ U “ H , then theclaim (4.31) follows from Lemmas 4.2 and 4.13.Suppose that r f n acts on U k for k ă l . To prove (4.31), it is enough to show that(4.32) r T “ r f n T . Indeed, if (4.32) holds, then by induction on the number of columns in T , we have r f n T “ ` U l , r T ˘ “ ´ U l , r f n T ¯ by (4.32) “ ´ U l , r f n T ¯ “ ´ U l , r f n T ¯ by induction hypothesis “ r f n ` U l , T ˘ by Lemma 4.13 , Hence we have r f n T “ r f n T which implies (4.31).Now we verify (4.32). Let us recall Table 1 and then we have(4.33) σ red “ r σ red . We will prove (4.32) for the case U tail k ‰ H with non-trivial sub-cases. The proofof other cases or the case U tail k “ H is almost identical, so we leave it to the readerto verify (4.32) for these cases.Let us recall the action of r f n in Section 4.5. Then for the case U tail k ‰ H , thesignature σ k ` (4.30) associated to U k ` must be ¨ or ` . Case 1.
If ht p U body k ` q ď ht p U body k q ă ht p U body k ´ q “ ht p U body k q `
2, then by definition of S k the top entry of U k ` can not be moved to the right in r T . Thus we obtain p σ k , σ k ` q “ p r σ k , r σ k ` q , which implies (4.32). Case 2.
If ht p U body k ` q “ ht p U body k q ` “ ht p U body k ´ q , then the signature σ k ` (4.30) cannot be ` . Otherwise p U k ` , U k ` q ć p U k , U k ´ q for Definition 3.2 (ii), which USZTIG DATA OF KN TABLEAUX IN TYPE D 29 is a contradiction. Thus σ k ` “ ¨ . We observe that p σ k , σ k ` q “ p` , ¨ q ÝÑ p r σ k , r σ k ` q “ p ¨ , `q ,U k ` r s “ n or n ´ , U k ` r s ě n ´ . (4.34) It is straightforward to check from the definition of S k that when we apply S k to T , the domino vd in U k is changed as follows:(4.35) $&% n ´ in U k is moved to n in U k ` below if U k ` r s “ n , n in U k is moved to n ´ in U k ` above if U k ` r s “ n ´ . Combining (4.33), (4.34) and (4.35), we conclude that (4.32) holds in thiscase.
Case 3.
If ht p U body k ` q ď ht p U body k q and ht p U body k q ` ă ht p U body k ´ q , then we have p U k ` , U k ` q Ÿ p U k , U k ´ q ÝÑ p U k ` , U k ` q Ÿ p U k , U k ´ q . By (4.33), this implies that (4.32) holds in this case.
Case 4.
Suppose ht p U body k ` q “ ht p U body k q ` p U body k q ` ă ht p U body k ´ q . First weconsider the case σ k ` “ ¨ . If there exists i such that U k ` p i q ą U k p i ` a k ´ q ,where a k “ ht p U ail k q , then by definition p U k ` , U k ` q Ž p U k , U k ´ q ÝÑ p U k ` , U k ` q Ž p U k , U k ´ q . Form this we observe that the domino vd in U k is not moved when we applythe sliding to T . Also we note that p σ k , σ k ` q “ p` , ¨ q ÝÑ p r σ k , r σ k ` q “ p` , ¨ q . Combining these observations with (4.33), we obtain (4.32) in this case. Ifthere is no such i , we have p U k ` , U k ` q Ž p U k , U k ´ q ÝÑ p U k ` , U k ` q Ÿ p U k , U k ´ q , p σ k , σ k ` q “ p` , ¨ q ÝÑ p r σ k , r σ k ` q “ p ¨ , `q . Note that r U k r s “ U k ` r s ď n ´ σ k ` “ ` . In this case, we obtain p U k ` , U k ` q Ž p U k , U k ´ q ÝÑ p U k ` , U k ` q Ž p U k , U k ´ q , since U k ` r s ą n ´
1. Also it is clear that p r σ k , r σ k ` q “ p` , `q by definitionof S k . On the other hand, the domino vd in U k can not be moved to the leftwhen we apply the sliding to T . Consequently, we have (4.32).We complete the proof of Theorem 4.12. (cid:3) Embedding into the crystal of Lusztig data
In this section, we describe the embedding of V λ into the crystal of Lusztig data.5.1. PBW crystal.
Let N “ n ´ n , which is the length of the longest element w P W of type D n . Let i “ p i , . . . , i N q be the reduced expression of w P W corresponding to the following convex ordering on Φ ` : ǫ i ` ǫ j ă ǫ k ´ ǫ l ,ǫ i ` ǫ j ă ǫ k ` ǫ l ðñ p j ą l q or p j “ l , i ą k q ,ǫ i ´ ǫ j ă ǫ k ´ ǫ l ðñ p i ă k q or p i “ k , j ă l q , (5.1)for 1 ď i ă j ď n and 1 ď k ă l ď n (see [5, Section 3.1]). We have(5.2) Φ ` “ t β : “ α i ă β : “ s i p α i q ă . . . ă β N : “ s i ¨ ¨ ¨ s i N ´ p α i N qu , where β “ α n . Let Φ ` J “ t ǫ i ´ ǫ j | ď i ă j ď n u be the set of positive roots of l and Φ ` p J q “ t ǫ i ` ǫ j | ď i ă j ď n u be the set of roots of the nilradical u of theparabolic subalgebra of g associated to l . Then we haveΦ ` p J q “ t β i | ď i ď M u , Φ J “ t β i | M ` ď i ď N u , where M “ N {
2. Let(5.3) B “ t c “ p c β , . . . , c β N q | c β i P Z ` p ď i ď N q u “ Z N ` . Then B becomes a crystal isomorphic to that the negative part of U q p g q , which iscalled the crystal of Lusztig data associated to i . Let B J “ c “ p c β q P B ˇˇ c β “ β P Φ ` p J q ( , B J “ c “ p c β q P B ˇˇ c β “ β P Φ J ( , (5.4)be the subcrystals of B , where we assume that r e n c “ r f n c “ with ε n p c q “ ϕ n p c q “´8 for c P B J . The crystal B J can be viewed as a crystal of a quantum nilpotentsubalgebra, while B J is isomorphic to the crystal of U ´ q p l q as an l -crystal.The reduced expression i enables us to describe the crystal structure on B veryexplicitly [5, Proposition 3.2]. In particular, we have Proposition 5.1. [5, Corollary 3.5]
The map (5.5) B J b B J / / Bc J b c J ✤ / / c is an isomorphism of crystals, where c is the concatenation of c J and c J given by c “ p c β , . . . , c β M loooooomoooooon c J , c β M ` , . . . , c β N loooooooomoooooooon c J q . USZTIG DATA OF KN TABLEAUX IN TYPE D 31
Remark 5.2.
One may find an explicit description of the reduced expression i “p i , . . . , i N q of w corresponding to (5.1) [5, Section 3.1]. Recall that there is aone-to-one correspondence between the reduced expressions of w and the convexorderings on Φ ` [18]. Then the subexpression p i , . . . , i M q corresponding to the rootsof u always appears as the first M entries (up to 2-term braid moves) in any reducedexpression of w such that the positive roots of u precede those of l with respectto the corresponding convex ordering. Here a 2-term braid move means ij “ ji for i, j P I such that | i ´ j | ą Remark 5.3.
In [5, Proposition 3.2], we give an explicit description of the operator r f i on B , which is obtained by applying [16, Theorem 4.5] to the reduced expression i (see also [5, Remark 3.3]). Remark 5.4.
There is another reduced expression i of w , which gives a nice com-binatorial description of the crystal B i of Lusztig data associated to i [16]. Further-more, there is an explicit construction of isomorphism from the crystal of marginallylarge tableaux of type D to B i [17], where a marginally large tableau is a combina-torial model for the crystal of the negative part of the quantized enveloping algebraof classical type [4] ( not the crystal of a highest weight module).We remark that the reduced expression i in [17] is not equal to i here (up to2-term braid moves). It would be interesting to see whether we have similar resultsas in [5] with respect to i .5.2. RSK of type D n . Let Ω be the set of biwords p a , b q such that(1) a “ a ¨ ¨ ¨ a r and b “ b ¨ ¨ ¨ b r are finite words in r n s for some r ě a i ă b i for 1 ď i ď r ,(3) p a , b q ď ¨ ¨ ¨ ď p a r , b r q ,where p a, b q ă p c, d q if and only if p a ă c q or ( a “ c and b ą d ) for p a, b q , p c, d q .One may naturally identify Ω with B J by regarding |t k | p a k , b k q “ p a, b q u| as themultiplicity of ǫ i ` ǫ j when p a, b q “ p j, i q with i ă j .There is an analogue of RSK due to [1], which gives a bijection(5.6) V / / B J T ✤ / / c J p T q . For the reader’s convenience, let us briefly recall the bijection (5.6). For i P r n s and T P SST r n s p λ π q with λ P P n , we denote by T Ð i the tableau obtained byapplying the Schensted’s column insertion of i into T in a reverse way starting fromthe rightmost column of T so that p T Ð i q P SST r n s p µ π q for some µ obtained byadding a box in a corner of λ .Now, let T P SST p δ π q Ă V be given. Then c J p T q is given by the following steps: (1) Let x be the smallest entry in T such that it is located at the leftmostamong such entries and let T be the tableau obtained from T by removing x .(2) Take y which is the entry in T below x . Let T be the tableau obtainedfrom T by applying the inverse of the Schensted’s column insertion to y .Then we obtain an entry z such that p T Ð z q “ T .(3) We apply the above steps to T : “ T instead of T . Then we denote by x and z the entries obtained from T and let T : “ T .(4) In general, repeat this process for T i ` : “ T i p i “ , , . . . q until there isno entries in T i ` , and let x i ` and z i ` be the entries from T i ` by thisprocess.(5) Finally we obtain a biword p a , b q P Ω given by ˜ ab ¸ “ ˜ x x . . . x ℓ z z . . . z ℓ ¸ , and define c J p T q P B J to be the one corresponding to p a , b q . Theorem 5.5. [5, Theorem 4.6]
The bijection in (5.6) is an isomorphism of crystals.
Embedding of KN λ into B. Let µ P P n be given and let ǫ µ “ ř ni “ µ i ǫ n ´ i .Consider the map(5.7) S µ / / B J b T ǫ µ T ✤ / / c J p T q b t ǫ µ , where c J p T q is the one such that the multiplicity c ǫ i ´ ǫ j is equal to the number of i ’sappearing in the p n ´ j ` q th row of T for 1 ď i ă j ď n . Proposition 5.6.
The map (5.7) is an embedding of crystals.
Proof.
It is a well-known fact that the map is an embedding of l -crystals (cf. [12]).By definition of r e n and r f n on S µ and B J , it becomes a morphism of g -crystals. (cid:3) Corollary 5.7.
For λ P P n , the map (5.8) V λ b T rω n / / B J b B J b T ω λ S b T b t rω n ✤ / / c J p S q b c J p T q b t ω λ , is an embedding of crystals, where r “ p λ, ω n q . (cid:3) We are now in a position to state the main result in this paper.
Theorem 5.8.
For λ P P n , we have an embedding of crystals given by USZTIG DATA OF KN TABLEAUX IN TYPE D 33 KN λ B b T ω λ T λ V λ b T rω n B J b B J b T ω λ (3.11) Ξ λ (4.29) (5.8) (5.5) Proof.
It follows from Theorems 3.9, 4.12, Proposition 5.1 and Corollary 5.7. (cid:3)
Example 5.9.
Set n “ λ “ ` , , , , ´ ˘ . Let T P KN λ be given by T “
234 55 45 1 1 . By (3.11) and (4.29) (see Example 4.9), we haveΨ λ p T q “
55 3 44 ...... ......
15 23 121 (4.29)
ÝÝÝÝÝÑ
545 3 ............ ............
Note that σ “ p ´ , ` , ` , ´ , ¨ q and σ “ p ´ , ¨ , ` , ¨ , ¨ q . Also σ red “ p ´ , ` , ¨ , ¨ , ¨ q , σ red “ p ´ , ¨ , ` , ¨ , ¨ q (cf. Lemma 4.13).Put T “ Ψ λ p T q . Then T body “
545 31 1 , T tail “ . Let us recall (5.3) and (5.4). For simplicity, we use the notation in [5, Section 3.2]for c J p T body q associated to Φ ` p J q with the convex order (5.1), that is, we identify p c β , . . . , c β q P B J with c β c β c β c β c β c β c β c β c β c β . Here β “ α . Similarly, we use the above notation for c J p T tail q with respect toΦ J and the convex order (5.1), that is, we identify p β , . . . , β q P B J with c β c β c β c β c β c β c β c β c β c β . Here β “ α , β “ α , β “ α and β “ α (cf. [5, Example 3.1]).Now we find c J p T body q by the steps (1)–(5) in Section 5.2 as follows. ÝÝÑ , ˜ ¸ ÝÝÑ , ˜ ¸ ÝÝÑ H , ˜ ¸ . Thus we have
545 31 1 (5.6)
ÝÝÝÝÝÑ
10 0 P B J (cf. Example [5, Example 4.5]). Next by definition (5.7), we have c J p T tail q asfollows. (5.7) ÝÝÝÝÝÑ
01 1 P B J . Hence we obtain the Lusztig data for the KN tableau T associated to i (cf.Proposition 5.1), that is,Ξ λ p T q “ p , , , , , , , , , , , , , , , , , , , q b t ω λ . USZTIG DATA OF KN TABLEAUX IN TYPE D 35
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