Littlewood Richardson coefficients for reflection groups Arkady - PowerPoint PPT Presentation

Littlewood Richardson coefficients for reflection groups Arkady Berenstein and Edward Richmond* University of British Columbia Joint Mathematical Meetings Boston January 7, 2012 Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 1 / 20

Preliminaries on flag varieties 1 Algebraic approach to Schubert Calculus 2 Statement of results 3 Examples 4 The Nil-Hecke ring 5 Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 2 / 20

Preliminaries on flag varieties Preliminaries: Schubert Calculus of G/B Let G be a Kac-Moody group over C (or a simple Lie group). Fix T ⊆ B ⊆ G a maximal torus and Borel subgroup of G . Let W := N ( T ) /T denote the Weyl group G . Let G/B be the flag variety (projective ind-variety). For any w ∈ W , we have the Schubert variety X w = BwB/B ⊆ G/B. Denote the cohomology class of X w by σ w ∈ H 2 ℓ ( w ) ( G/B ) . Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 3 / 20

Preliminaries on flag varieties � Additively, we have that H ∗ ( G/B ) ≃ Z σ w . w ∈ W Goal (Schubert Calculus) Compute the structure (Littlewood-Richardson) coefficients c w u,v with respect to the Schubert basis defined by the product � c w σ u · σ v = u,v σ w . w ∈ W Note that if ℓ ( w ) � = ℓ ( u ) + ℓ ( v ) , then c w u,v = 0 . For any w, u, v ∈ W , we have that c w u,v ≥ 0 . (proofs are geometric) For example, if G is a finite Lie group, then the cardinality | g 1 X u ∩ g 2 X v ∩ g 3 X w 0 w | = c w u,v for a generic choice of g 1 , g 2 , g 3 ∈ G. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 4 / 20

Algebraic approach to Schubert Calculus Algebraic approach to Schubert calculus Let A = A ( G ) denote the Cartan matrix of G. Alternatively, fix a finite index set I and let A = { a ij } be an I × I matrix such that a ii = 2 a ij ∈ Z ≤ 0 if i � = j a ij = 0 ⇔ a ji = 0 . The matrix A defines an action of a Coxeter group W generated by reflections { s i } i ∈ I on the vector space V := Span C { α i } i ∈ I given by s i ( v ) := v − � v, α ∨ i � α i where � α i , α ∨ j � := a ij . In particular, Coxeter groups of this type are crystallographic. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 5 / 20

Algebraic approach to Schubert Calculus Algebraic approach to Schubert calculus If we abandon the group G , we can consider a matrix A as follows: Fix a finite index set I and let A = { a ij } be an I × I matrix such that a ii = 2 a ij ∈ R ≤ 0 if i � = j a ij = 0 ⇔ a ji = 0 . The matrix A defines an action of a Coxeter group W generated by reflections { s i } i ∈ I on the vector space V := Span C { α i } i ∈ I given by s i ( v ) := v − � v, α ∨ i � α i where � α i , α ∨ j � := a ij . Every Coxeter group can be represented as above. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 6 / 20

Algebraic approach to Schubert Calculus Some examples G = SL (4) (type A 3 )   − 1 2 0   and W = S 4 (symmetric group) − 1 − 1 A = 2 − 1 0 2 G = Sp (4) (type C 2 ) � � 2 − 2 A = and W = I 2 (4) (dihedral group of 8 elements) − 1 2 G = � SL (2) (affine type A ) � � 2 − 2 A = and W = I 2 ( ∞ ) (free dihedral group) − 2 2 Let ρ = 2 cos( π/ 5) � � − ρ 2 A = and W = I 2 (5) (dihedral group of 10 elements) − ρ 2 Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 7 / 20

Algebraic approach to Schubert Calculus Notation on sequences and subsets Any sequence i := ( i 1 , . . . , i m ) ∈ I m has a corresponding element s i 1 · · · s i m ∈ W. If s i 1 · · · s i m is a reduced word of some w ∈ W , then we say that i ∈ R ( w ) , the collection of reduced words. For each i ∈ I m and subset K = { k 1 < k 2 < · · · < k n } of the interval [ m ] := { 1 , 2 , . . . , m } let the subsequence i K := ( i k 1 , . . . , i k n ) ∈ I n . We say a sequence i is admissible if i j � = i j +1 for all j ∈ [ m − 1] . Observe that any reduced sequence is admissible. (In general, the converse is false.) Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 8 / 20

Algebraic approach to Schubert Calculus Definition Let m > 0 and let K, L be subsets of [ m ] := { 1 , 2 , . . . , m } such that | K | + | L | = m . We say that a bijection φ : K → [ m ] \ L is bounded if φ ( k ) < k for each k ∈ K . Definition Given a reduced sequence i = ( i 1 , . . . , i m ) ∈ I m , we say that a bounded bijection φ : K → [ m ] \ L is i - admissible if the sequence i L and the sequences i L ( k ) are admissible for all k ∈ K where L ( k ) := L ∪ φ ( K ≤ k ) . Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 9 / 20

Statement of results Recall that the Littlewood-Richardson coefficients c w u,v are defined by the product � c w σ u · σ v = u,v σ w . w ∈ W Theorem: Berenstein-R, 2010 Let u, v, w ∈ W such that ℓ ( w ) = ℓ ( u ) + ℓ ( v ) and let i = ( i 1 , . . . , i m ) ∈ R ( w ) . Then � c w u,v = p φ where the summation is over all triples (ˆ u, ˆ v, φ ) , where u, ˆ ˆ v ⊂ [ m ] such that i ˆ u ∈ R ( u ) , i ˆ v ∈ R ( v ) . φ : ˆ u ∩ ˆ v → [ m ] \ (ˆ u ∪ ˆ v ) is an i -admissible bounded bijection. The Theorem is still true without i -admissible. The Theorem generalizes to structure coefficients of T -equivariant cohomology H ∗ T ( G/B ) . Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 10 / 20

Statement of results Definition For any k ∈ [ m ] and i = ( i 1 , . . . , i m ) , denote α k := α i k and s k := s i k . For any bounded bijection φ : K → [ m ] \ L we define the monomial p φ ∈ Z by the formula − → p φ := ( − 1) | K | � � � w k ( α k ) , α ∨ φ ( k ) � where w k := s r k ∈ K r ∈ L ( k ) φ ( k ) <r<k − � is taken in the natural order induced by the sequence [ m ] → where the product and if the product is empty, we set w k = 1 . Also, if K = ∅ , then p φ = 1 . Theorem: Berenstein-R, 2010 If the Cartan matrix A = ( a ij ) satisfies a ij · a ji ≥ 4 ∀ i � = j, then p φ ≥ 0 when φ is an i -admissible bounded bijection. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 11 / 20

Example: G = � Examples SL (2) Examples where G = � SL (2) , i = (1 , 2 , 1 , 2) � � − 2 2 A = − 2 2 Compute c w u,v where w = s 1 s 2 s 1 s 2 , u = v = s 1 s 2 . Find ˆ u, ˆ v ⊆ [4] = { 1 , 2 , 3 , 4 } 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 For 1 2 3 4 , we have u ∩ ˆ v = { 3 , 4 } [4] \ (ˆ u ∪ ˆ v ) = { 1 , 2 } ˆ and with bounded bijections φ 1 : (3 , 4) → (1 , 2) φ 2 : (3 , 4) → (2 , 1) . NOTE: φ 1 is not i -admissible. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 12 / 20

Example: G = � Examples SL (2) Examples where G = � SL (2) , i = (1 , 2 , 1 , 2) For 1 2 3 4 , we have bounded bijections φ 1 : (3 , 4) → (1 , 2) φ 2 : (3 , 4) → (2 , 1) . p φ 1 = � α 3 , α ∨ 1 � · � s 3 ( α 4 ) , α ∨ p φ 2 = � α 3 , α ∨ 2 � · � s 2 s 3 ( α 4 ) , α ∨ 2 � 1 � Totaling over all bounded bijections, we have ❤❤❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭✭✭ c w � α 3 , α ∨ 1 � · � s 3 ( α 4 ) , α ∨ 2 � + � α 3 , α ∨ 2 � · � s 2 s 3 ( α 4 ) , α ∨ 1 � = ❤ u,v ❳❳❳❳❳ ✘ ❳❳❳❳❳ ✘ − ✘✘✘✘✘ 2 � − ✘✘✘✘✘ � s 3 ( α 4 ) , α ∨ � s 3 ( α 4 ) , α ∨ 2 � + 1 + 1 ❳ ❳ 4 + 4 + ✁ 2 + ✁ − ✁ ❆ ❆ ❆ = 2 + 1 + 1 = 6 With only i -admissible terms. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 13 / 20

Example: G = � Examples SL (2) Examples where G = � SL (2) , i = (1 , 2 , 1 , 2 , 1) Compute c w u,v where w = s 1 s 2 s 1 s 2 s 1 , u = s 1 s 2 s 1 , v = s 2 s 1 . Find ˆ u, ˆ v ⊆ [5] = { 1 , 2 , 3 , 4 , 5 } 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Totaling over all bounded bijections, we have ❤❤❤❤❤❤❤❤❤ ✭ ❤❤❤❤❤❤❤❤❤❤ ✭ 3 � + ✭✭✭✭✭✭✭✭✭✭ ✭✭✭✭✭✭✭✭✭ � α 4 , α ∨ 2 � · � s 4 ( α 5 ) , α ∨ � α 4 , α ∨ 3 � · � s 3 s 4 ( α 5 ) , α ∨ c w = 2 � ❤ ❤ u,v ❤❤❤❤❤❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭✭✭✭✭✭ � s 3 ( α 4 ) , α ∨ 1 � · � s 3 s 4 ( α 5 ) , α ∨ 2 � + � s 3 ( α 4 ) , α ∨ 2 � · � s 2 s 3 s 4 ( α 5 ) , α ∨ + 1 � ❤ ❳❳❳❳❳ ✘ ❳❳❳❳❳ ✘ 1 � − ✘✘✘✘✘ 3 � − ✘✘✘✘✘ � s 2 ( α 3 ) , α ∨ � s 4 ( α 5 ) , α ∨ � s 4 ( α 5 ) , α ∨ 3 � − � s 2 s 3 s 4 ( α 5 ) , α ∨ − 1 � ❳ ❳ + 1 + 1 4 + ✁ 4 − ✁ 4 + 4 + 2 + ✁ 2 + ✁ − ✁ ❆ ❆ ❆ ❆ ❆ = 2+2 + 1 + 1 = 10 With only i -admissible terms. Arkady Berenstein and Edward Richmond* (UBC) L-R coefficients January 7, 2012 14 / 20