Combinatorics of generalized exponents edric Lecouvey and Cristian - PowerPoint PPT Presentation

Combinatorics of generalized exponents edric Lecouvey and Cristian Lenart † C´ University of Tours, France; State University of New York at Albany, USA † FPSAC 2019 University of Ljubljana, Slovenia International Mathematics Research Notices , 2018, DOI 10.1093/imrn/rny157 Cristian Lenart was partially supported by the NSF grant DMS–1362627.

Representations of semisimple Lie algebras Consider a complex semisimple Lie algebra g . ◮ R = R + ⊔ R − root system, ◮ P weight lattice, ◮ P + dominant weights, ◮ ω i fundamental weights ( i ∈ I ), ◮ W Weyl group. Type A n − 1 : ◮ g = sl n , ◮ weights are compositions, ◮ dominant weights are partitions (Young diagrams), ◮ ω i = (1 i ), ◮ W = S n .

Representations of semisimple Lie algebras (cont.) For a dominant weight λ ∈ P + , let V ( λ ) be the irreducible representation with highest weight λ , and P ( λ ) its weights. In classical types, a basis of V ( λ ) is indexed by Kashiwara-Nakashima tableaux and King tableaux of shape λ . Type A n − 1 : semistandard Young tableaux (SSYT). 1 2 2 3 T = λ = (4 , 2 , 1) , weight ( T ) = (1 , 3 , 1 , 2) . 2 4 4

Lusztig’s t -analogue of weight multiplicity For µ ∈ P ( λ ), let K λ,µ be the multiplicity of µ in V ( λ ). (In type A , this is the number of SSYT of shape λ , weight µ .) Lusztig defined the t -analogue K λ,µ ( t ), i.e., K λ,µ (1) = K λ,µ , via w ∈ W sgn ( w ) x w ( λ + ρ ) − ρ � K λ,µ ( t ) x µ . � = � α ∈ R + (1 − tx − α ) µ ∈ P ( λ )

Importance of K λ,µ ( t ) K λ,µ ( t ), for λ, µ dominant, is also known as a Kostka-Foulkes polynomial. This polynomial has remarkable properties: ◮ it is a special affine Kazhdan-Lusztig polynomial, so K λ,µ ( t ) ∈ Z ≥ 0 [ t ]; ◮ it records the Brylinski-Kostant filtration of the µ -weight space V ( λ ) µ ; ◮ it is related to Hall-Littlewood polynomials (i.e., specializations of Macdonald polynomials at q = 0): � s λ ( x ) = K λ,µ ( t ) P µ ( x ; t ) , µ ∈ P + where s λ ( x ) are the Weyl characters (Schur polynomials in type A ).

Combinatorial formulas In type A n − 1 , K λ,µ ( t ) is expressed combinatorially via the Lascoux-Sch¨ utzenberger charge statistic on SSYT. Finding combinatorial formulas beyond type A has been a long-standing problem. Goal. The first such formula, for K λ, 0 ( t ) in type C n ( g = sp 2 n ). We also have: related formulas, applications, as well as the possibility to extend to all K λ,µ ( t ) and types B , D . Remark. The special case µ = 0 is, in fact, the most complex one. Kostant called K λ, 0 ( t ) generalized exponents, as the classical ones are obtained when λ is the highest root. Approach. Extend another combinatorial formula in type A , due to Lascoux-Leclerc-Thibon (LLT), which is based on Kashiwara’s crystal graphs; our approach is simpler compared to LLT.

Kashiwara’s crystal graphs Encode irreducible representations V ( λ ) of the corresponding quantum group U q ( g ) as q → 0. Kashiwara (crystal) operators are modified versions of the Chevalley generators: e i , f i , i ∈ I . Fact. V ( λ ) has a crystal basis B ( λ ): in the limit q → 0 we have f i , e i : B ( λ ) → B ( λ ) ⊔ { 0 } , f i ( b ) = b ′ e i ( b ′ ) = b . ⇐ ⇒ Encode as colored directed graph: → b ′ . i f i ( b ) = b ′ ⇐ ⇒ b − Fact. Classical crystals are realized as graphs on Kashiwara-Nakashima tableaux.

Example. g = sl 4 , λ = (3 , 3 , 1), blue: α 1 = ε 1 − ε 2 , green: α 2 = ε 2 − ε 3 , red: α 3 = ε 3 − ε 4 .

The LLT formula Notation. ε i ( b ) = max { k : e k ϕ i ( b ) = max { k : f k i ( b ) � = 0 } , i ( b ) � = 0 } , � � ε ( b ) := ε i ( b ) ω i , | ε ( b ) | = i ε i ( b ) , ϕ ( b ) , | ϕ ( b ) | . i ∈ I i ∈ I Theorem. [Lascoux, Leclerc, Thibon] In type A n − 1 , we have t | ε ( b ) | . � K λ, 0 ( t ) = b ∈ B ( λ ) 0 There is a more involved formula for the other K λ,µ ( t ).

Our approach to K λ, 0 ( t ) in classical types Notation. ◮ P and P n denote all partitions and partitions with at most n parts; ◮ P (2) denotes partitions with all parts/rows even; ◮ P (1 , 1) denotes partitions with all columns of even height; ◮ c λ ν ( sp 2 n ) is the branching coefficient for the restriction from gl 2 n to sp 2 n , corresponding to the weights ν ∈ P 2 n and λ ∈ P n , respectively. By classical results (Kostant, Hesselink, Littlewood), we derive in type C n (and similarly in the other classical types): K C n λ, 0 ( t ) t | ν | / 2 c λ � i =1 (1 − t 2 i ) = ν ( sp 2 n ) . � n ν ∈P (2) 2 n

Other ingredients ◮ the stable branching rule c λ � c ν ν ( sp ∞ ) = λ,δ , δ ∈P (1 , 1) where c ν λ,δ are the (type A ) Littlewood-Richardson coefficients, giving the multiplicity of V ( ν ) in V ( λ ) ⊗ V ( δ ); ◮ the combinatorial formula for c ν λ,δ in terms of the crystal: c ν λ,δ = | LR ν λ,δ | , where LR ν λ,δ = { b ∈ B ( λ ) : ε ( b ) ≤ δ , ϕ ( b ) = ε ( b ) + ν − δ } .

Immediate consequences ◮ new short proof of the LLT formula in type A ; ◮ stable versions K X ∞ λ, 0 ( t ) of K X n λ, 0 ( t ) when the rank n goes to ∞ , for X ∈ { A , B , C , D } . Remark. We have K B ∞ λ, 0 ( t ) = K D ∞ K B ∞ λ, 0 ( t ) = K C ∞ λ, 0 ( t ) , λ ′ , 0 ( t ) .

Ingredients for finite rank: type C n ◮ a nonstable stable branching rule expressing c λ ν ( sp 2 n ) outside the stable range ν ∈ P n , namely when ν ∈ P 2 n \ P n ; based on recent work of J.-H. Kwon on his spin model for symplectic crystals; ◮ one of many versions of the combinatorial map expressing the symmetry of LR coefficients: λ,δ = c ν ′ c ν λ ′ ,δ ′ .

The nonstable branching rule Fix λ ∈ P n . Recall that when ν ∈ P n (stable case), we have c λ � c ν ν ( sp 2 n ) = λ,δ , δ ∈P (1 , 1) 2 n λ,δ | = | LR ν ′ where c ν λ,δ = | LR ν λ ′ ,δ ′ | . But this fails for general ν ∈ P 2 n . Theorem. [Lecouvey, L.; based on Kwon] For ν ∈ P 2 n , we have c λ � c ν ν ( sp 2 n ) = λ,δ , δ ∈P (1 , 1) 2 n where λ,δ = |{ T ∈ LR ν ′ c ν λ ′ ,δ ′ : r i > δ rev 2 i − 1 = δ rev 2 i }| , and ( r 1 ≤ . . . ≤ r p ) is the first row of T .

The formula for K C n λ, 0 ( t ) Notation. D 2 n ( λ ) denotes the subset of distinguished vertices in B 2 n ( λ ) of type A 2 n − 1 , that is, vertices b with ◮ ϕ i ( b ) = 0 for any odd i , ◮ ε i ( b ) even for any odd i ; ◮ flag condition: the entries in row i are ≥ 2 i − 1. Main theorem. [Lecouvey, L.] We have t | ε ∗ ( b )+ µ b , n | / 2 . � K C n λ, 0 ( t ) = b ∈ D 2 n ( λ ) where 2 n − 1 � ε i ( b ) � � | ε ∗ ( b ) + µ b , n | / 2 = (2 n − i ) . 2 i =1

Another version of the formula Goal. Express K C n λ, 0 ( t ) in terms a combinatorial set naturally indexing a basis of the 0-weight space V ( λ ) 0 . Definition. King tableaux are SSYT of a given shape λ in the alphabet { 1 < 1 < 2 < 2 < . . . < n < n } satisfying: the entries in row i are ≥ i . Fact. There is an easy bijection between D 2 n ( λ ) and King tableaux.

Applications of our formula for K C n λ, 0 ( t ) ◮ K C n +1 λ, 0 ( t ) − K C n λ, 0 ( t ) ∈ Z ≥ 0 [ t ]; ω 2 p , 0 ( t ) = K A n − 1 ◮ K C n γ p , 0 ( t 2 ), where γ p = (2 p , 1 n − 2 p ) (conjectured by Lecouvey); ◮ calculation of the smallest power in K C n λ, 0 ( t ).

Next goal Extend our work from K C n λ, 0 ( t ) to all K C n λ,µ ( t ). Main idea. Extend the statistic on vertices of weight 0 to the whole crystal via an atomic decomposition of the crystal; see our poster.