Separable elements in Weyl groups Yibo Gao Joint work with: - PowerPoint PPT Presentation

Separable elements in Weyl groups Yibo Gao Joint work with: Christian Gaetz Massachusetts Institute of Technology Summer Combo in Vermont, 2019 Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 1 / 23

Overview Background 1 separable permutations results by Fan Wei root systems and Weyl groups Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 2 / 23

Overview Background 1 separable permutations results by Fan Wei root systems and Weyl groups Separable elements in Weyl groups 2 definition properties Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 2 / 23

Overview Background 1 separable permutations results by Fan Wei root systems and Weyl groups Separable elements in Weyl groups 2 definition properties Classification via pattern avoidance 3 Case 1: Simply-laced Case 2: Type B n , C n Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 2 / 23

Separable permutations Definition A permutation is separable if it avoids the patterns 3142 and 2413. Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 3 / 23

Separable permutations Definition A permutation is separable if it avoids the patterns 3142 and 2413. • • • • • • • • Figure: Permutations 3142 and 2413. Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 3 / 23

Separable permutations Definition A permutation is separable if it avoids the patterns 3142 and 2413. • • • • • • • • Figure: Permutations 3142 and 2413. Lemma If w ∈ S n is separable, then there exists 1 < m < n such that either w 1 · · · w m is a separable permutation on { 1 , . . . , m } and w m +1 · · · w n is a separable permutation on { m + 1 , . . . , n } ; or w 1 · · · w n − m is a separable permutation on { m + 1 , . . . , n } and w n − m +1 · · · w n is a separable permutation on { 1 , . . . , m } . Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 3 / 23

Separable permutations: fun facts (Wikipedia) Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 4 / 23

Separable permutations: fun facts (Wikipedia) Separable permutations were first introduced by Bose, Buss and Lubiw in 1998 via a structure of rooted binary tree structure. They gave characterizations using pattern avoidance as well. Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 4 / 23

Separable permutations: fun facts (Wikipedia) Separable permutations were first introduced by Bose, Buss and Lubiw in 1998 via a structure of rooted binary tree structure. They gave characterizations using pattern avoidance as well. Separable permutations are counted by Schr¨ oder numbers. • • • • • • • • • Figure: A Schr¨ oder path: lattice path from (0 , 0) to (2 n , 0) using steps (1 , 1), (1 , − 1), (2 , 0) that never goes below the x -axis. Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 4 / 23

Separable permutations: fun facts (Wikipedia) Separable permutations were first introduced by Bose, Buss and Lubiw in 1998 via a structure of rooted binary tree structure. They gave characterizations using pattern avoidance as well. Separable permutations are counted by Schr¨ oder numbers. • • • • • • • • • Figure: A Schr¨ oder path: lattice path from (0 , 0) to (2 n , 0) using steps (1 , 1), (1 , − 1), (2 , 0) that never goes below the x -axis. If a collection of distinct real polynomials all have equal values at some number x, then the permutation that describes how the numerical ordering of the polynomials changes at x is separable, and every separable permutation can be realized in this way. Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 4 / 23

Notations on ranked posets Let P be a finite ranked poset with rank decomposition P 0 ⊔ P 1 ⊔ · · · ⊔ P r . Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 5 / 23

Notations on ranked posets Let P be a finite ranked poset with rank decomposition P 0 ⊔ P 1 ⊔ · · · ⊔ P r . We say that P is rank symmetric if | P i | = | P r − i | for all i , rank unimodal if there exists m such that | P 0 | ≤ | P 1 | ≤ · · · ≤ | P m | ≥ · · · ≥ | P r − 1 | ≥ | P r | . Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 5 / 23

Notations on ranked posets Let P be a finite ranked poset with rank decomposition P 0 ⊔ P 1 ⊔ · · · ⊔ P r . We say that P is rank symmetric if | P i | = | P r − i | for all i , rank unimodal if there exists m such that | P 0 | ≤ | P 1 | ≤ · · · ≤ | P m | ≥ · · · ≥ | P r − 1 | ≥ | P r | . For x ∈ P , let V x := { y ∈ P : y ≥ x } be the principal upper order ideal at x , Λ x := { y ∈ P : y ≤ x } be the principal lower order ideal at x . Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 5 / 23

Notations on ranked posets Let P be a finite ranked poset with rank decomposition P 0 ⊔ P 1 ⊔ · · · ⊔ P r . We say that P is rank symmetric if | P i | = | P r − i | for all i , rank unimodal if there exists m such that | P 0 | ≤ | P 1 | ≤ · · · ≤ | P m | ≥ · · · ≥ | P r − 1 | ≥ | P r | . For x ∈ P , let V x := { y ∈ P : y ≥ x } be the principal upper order ideal at x , Λ x := { y ∈ P : y ≤ x } be the principal lower order ideal at x . Let � q rk ( x ) F ( P ) = F ( P , q ) := x ∈ P be the rank generating function of P . Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 5 / 23

Background on the weak (Bruhat) order The right weak (Bruhat) order R n is generated by w ⋖ R ws i if ℓ ( ws i ) = ℓ ( w ) + 1 , where s i = ( i , i + 1) . Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 6 / 23

Background on the weak (Bruhat) order The right weak (Bruhat) order R n is generated by w ⋖ R ws i if ℓ ( ws i ) = ℓ ( w ) + 1 , where s i = ( i , i + 1) . The left weak (Bruhat) order L n is generated by w ⋖ L s i w if ℓ ( s i w ) = ℓ ( w ) + 1 , where s i = ( i , i + 1) . Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 6 / 23

Background on the weak (Bruhat) order The right weak (Bruhat) order R n is generated by w ⋖ R ws i if ℓ ( ws i ) = ℓ ( w ) + 1 , where s i = ( i , i + 1) . The left weak (Bruhat) order L n is generated by w ⋖ L s i w if ℓ ( s i w ) = ℓ ( w ) + 1 , where s i = ( i , i + 1) . • 321 • 321 • • 312 • • 312 231 231 • • 132 • • 132 213 213 • 123 • 123 Figure: The left weak order and the right weak order on S 3 . Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 6 / 23

Results by Fan Wei Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 7 / 23

Results by Fan Wei Theorem (Wei 2012) Let π ∈ S n be a separable permutation. Then both Λ π and V π are rank symmetric and rank unimodal. Moreover, F (Λ π ) F ( V π ) = F ( S n ) . Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 7 / 23

Results by Fan Wei Theorem (Wei 2012) Let π ∈ S n be a separable permutation. Then both Λ π and V π are rank symmetric and rank unimodal. Moreover, F (Λ π ) F ( V π ) = F ( S n ) . Her proof relies on the following lemma. Lemma (Wei 2012) Let π = uv as words where u and v are separable. Then if u ∈ S 1 ,..., m , v ∈ S m +1 ,..., n , F (Λ π ) = F (Λ u ) F (Λ v ) and � n � F ( V π ) = F ( V u ) F ( V v ) q ; m � n � if u ∈ S m +1 ,..., n , v ∈ S 1 ,..., m , F (Λ π ) = F (Λ u ) F (Λ v ) q and m F ( V π ) = F ( V u ) F ( V v ) . Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 7 / 23

Results by Fan Wei Theorem (Wei 2012) Let π ∈ S n be a separable permutation. Then both Λ π and V π are rank symmetric and rank unimodal. Moreover, F (Λ π ) F ( V π ) = F ( S n ) . Her proof relies on the following lemma. Lemma (Wei 2012) Let π = uv as words where u and v are separable. Then if u ∈ S 1 ,..., m , v ∈ S m +1 ,..., n , F (Λ π ) = F (Λ u ) F (Λ v ) and � n � F ( V π ) = F ( V u ) F ( V v ) q ; m � n � if u ∈ S m +1 ,..., n , v ∈ S 1 ,..., m , F (Λ π ) = F (Λ u ) F (Λ v ) q and m F ( V π ) = F ( V u ) F ( V v ) . We will be generalizing these results to other types. Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 7 / 23

Root systems and Weyl groups Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 8 / 23

Root systems and Weyl groups Definition (Root system) Let E = R n . A root system Φ ⊂ E is a finite set of vectors, such that Φ spans E ; for α ∈ Φ, k α ∈ Φ iff k ∈ {± 1 } ; for α, β ∈ Φ, 2( α, β ) / ( α, α ) ∈ Z ; for α, β ∈ Φ, σ α ( β ) := β − 2 � � α ∈ Φ. ( α, β ) / ( α, α ) Yibo Gao (MIT) Separable elements in Weyl groups Summer Combo 2019 8 / 23