Cluster Structures of Double Bott-Samelson Cells Daping Weng - PowerPoint PPT Presentation

Cluster Structures of Double Bott-Samelson Cells Cluster Structures of Double Bott-Samelson Cells Daping Weng Michigan State University April 2019 Joint work with Linhui Shen arXiv:1904.07992

Cluster Structures of Double Bott-Samelson Cells Motivation: Bott-Samelson Variety Let G, B, W be defined as usual. Let i = ( i 1 , . . . , i l ) be a reduced word of w . The Bott-Samelson variety associated to the reduced word i is � P i 1 × B P i 2 × B . . . × B P i l B where P i = B ⊔ B s i B.

Cluster Structures of Double Bott-Samelson Cells Motivation: Bott-Samelson Variety Let G, B, W be defined as usual. Let i = ( i 1 , . . . , i l ) be a reduced word of w . The Bott-Samelson variety associated to the reduced word i is � P i 1 × B P i 2 × B . . . × B P i l B where P i = B ⊔ B s i B. Note that � P i 1 × B . . . × B P i l = (B s j 1 B) × B . . . × B (B s j m B) j ⊂ i where j = ( j 1 , . . . , j m ) runs over all subwords of i (not necessarily reduced). These can be thought of as “Bott-Samelson cell”.

Cluster Structures of Double Bott-Samelson Cells Motivation: Bott-Samelson Variety Let G, B, W be defined as usual. Let i = ( i 1 , . . . , i l ) be a reduced word of w . The Bott-Samelson variety associated to the reduced word i is � P i 1 × B P i 2 × B . . . × B P i l B where P i = B ⊔ B s i B. Note that � P i 1 × B . . . × B P i l = (B s j 1 B) × B . . . × B (B s j m B) j ⊂ i where j = ( j 1 , . . . , j m ) runs over all subwords of i (not necessarily reduced). These can be thought of as “Bott-Samelson cell”. Alternatively one can think of an element of (B s j 1 B) × B . . . × B (B s j m B) as a sequence of flags that satisfies the relative position conditions imposed by the simple reflections s j 1 , s j 2 , . . . , s j m . So a “double Bott-Samelson cell” will then be two sequences of flags that satisfy two sequences of relative position conditions imposed by two words i and j .

Cluster Structures of Double Bott-Samelson Cells Definition Let G be the Kac-Peterson group (the smallest Kac-Moody group) associated to a symmetrizable generalized Cartan matrix and let B ± be the two opposite Borel subgroups.

Cluster Structures of Double Bott-Samelson Cells Definition Let G be the Kac-Peterson group (the smallest Kac-Moody group) associated to a symmetrizable generalized Cartan matrix and let B ± be the two opposite Borel subgroups. Let B ± = { Borel subgroups that are conjugates of B ± } . Bruhat decomposition implies that the G-orbits in B + × B + and B − × B − are parametrized by the Weyl group W.

Cluster Structures of Double Bott-Samelson Cells Definition Let G be the Kac-Peterson group (the smallest Kac-Moody group) associated to a symmetrizable generalized Cartan matrix and let B ± be the two opposite Borel subgroups. Let B ± = { Borel subgroups that are conjugates of B ± } . Bruhat decomposition implies that the G-orbits in B + × B + and B − × B − are parametrized by the Weyl group W. Notation We use superscript to denote Borel subgroups in B + , e.g. B 0 , B 1 , etc. We use subscript to denote Borel subgroups in B − , e.g. B 0 , B 1 , etc. � B 1 if w We write B 0 B 0 , B 1 � � is in the w -orbit in B + × B + . w � B 1 if (B 0 , B 1 ) is in the w -orbit in B − × B − . We write B 0 B 0 if B 0 , B 0 � � We write B 0 = ( g B − , g B + ) for some g ∈ G.

Cluster Structures of Double Bott-Samelson Cells Definition Let G be the Kac-Peterson group (the smallest Kac-Moody group) associated to a symmetrizable generalized Cartan matrix and let B ± be the two opposite Borel subgroups. Let B ± = { Borel subgroups that are conjugates of B ± } . Bruhat decomposition implies that the G-orbits in B + × B + and B − × B − are parametrized by the Weyl group W. Notation We use superscript to denote Borel subgroups in B + , e.g. B 0 , B 1 , etc. We use subscript to denote Borel subgroups in B − , e.g. B 0 , B 1 , etc. � B 1 if w We write B 0 B 0 , B 1 � � is in the w -orbit in B + × B + . w � B 1 if (B 0 , B 1 ) is in the w -orbit in B − × B − . We write B 0 B 0 if B 0 , B 0 � � We write B 0 = ( g B − , g B + ) for some g ∈ G. B + s i B + / B + can be thought of as the moduli space of B 1 satisfying s i � B 1 for a fixed B 0 . B 0

Cluster Structures of Double Bott-Samelson Cells Definition Definition Let b and d be two positive braids in the associated braid group. First choose a word ( i 1 , i 2 , . . . , i m ) for b and a word ( j 1 , j 2 , . . . , j n ) for d . The undecorated double Bott-Samelson cell Conf b d ( B ) is defined to be   B 0 s i 1 � B 1 s i 2 � . . . s im � B m   �     G � B 1 s j 2 � . . . s jn � B n B 0 s j 1      

Cluster Structures of Double Bott-Samelson Cells Definition Definition Let b and d be two positive braids in the associated braid group. First choose a word ( i 1 , i 2 , . . . , i m ) for b and a word ( j 1 , j 2 , . . . , j n ) for d . The undecorated double Bott-Samelson cell Conf b d ( B ) is defined to be   B 0 s i 1 � B 1 s i 2 � . . . s im � B m   �     G � B 1 s j 2 � . . . s jn � B n B 0 s j 1       Remark The resulting space does not depend on the choice of words for b and d .

Cluster Structures of Double Bott-Samelson Cells Definition Let U ± := [B ± , B ± ] and define decorated flag varieties A ± := G / U ± . We denote decorated flags with a symbol A instead of B. Definition The decorated double Bott-Samelson cell Conf b d ( A ) is defined to be   A 0 s i 1 � B 1 s i 2 � . . . s im � B m   �     G � B 1 s j 2 � . . . s jn � A n B 0 s j 1      

Cluster Structures of Double Bott-Samelson Cells Definition Let U ± := [B ± , B ± ] and define decorated flag varieties A ± := G / U ± . We denote decorated flags with a symbol A instead of B. Definition The decorated double Bott-Samelson cell Conf b d ( A ) is defined to be   A 0 s i 1 � B 1 s i 2 � . . . s im � B m   �     G � B 1 s j 2 � . . . s jn � A n B 0 s j 1       Decorated double Bott-Samelson cell can be viewed as a generalization of double Bruhat cells B + u B + ∩ B − v B − . Double Bruhat cells are examples of cluster varieties and are studied by Berenstein, Fomin, and Zelevinsky [BFZ05], Fock and Goncharov [FG06], and many others.

Cluster Structures of Double Bott-Samelson Cells Definition Let U ± := [B ± , B ± ] and define decorated flag varieties A ± := G / U ± . We denote decorated flags with a symbol A instead of B. Definition The decorated double Bott-Samelson cell Conf b d ( A ) is defined to be   A 0 s i 1 � B 1 s i 2 � . . . s im � B m   �     G � B 1 s j 2 � . . . s jn � A n B 0 s j 1       Decorated double Bott-Samelson cell can be viewed as a generalization of double Bruhat cells B + u B + ∩ B − v B − . Double Bruhat cells are examples of cluster varieties and are studied by Berenstein, Fomin, and Zelevinsky [BFZ05], Fock and Goncharov [FG06], and many others. Theorem (Shen-W.) The decorated double Bott-Samelson cells Conf b d ( A ) are smooth affine varieties.

Cluster Structures of Double Bott-Samelson Cells Cluster Structures We equip each double Bott-Samelson cell (both undecorated and decorated) with an atlas of algebraic torus charts, parametrized by a choice of words for b and d and a triangulation of the “trapezoid”. s 1 � B 1 s 1 � B 2 s 3 � B 3 B 0 s 2 � B 1 s 1 � B 2 s 2 � B 3 s 3 � B 4 s 3 � B 5 B 0

Cluster Structures of Double Bott-Samelson Cells Cluster Structures We equip each double Bott-Samelson cell (both undecorated and decorated) with an atlas of algebraic torus charts, parametrized by a choice of words for b and d and a triangulation of the “trapezoid”. s 1 � B 1 s 1 � B 2 s 3 � B 3 B 0 s 2 � B 1 s 1 � B 2 s 2 � B 3 s 3 � B 4 s 3 � B 5 B 0 There are two kinds of moves available to us: s 1 � B 1 s 1 � B 2 s 3 � B 3 B 0 diagonal flipping s 2 � B 1 s 1 � B 2 s 2 � B 3 s 3 � B 4 s 3 � B 5 B 0 s 1 � B 1 s 1 � B 2 s 3 � B 3 B 0 Braid move s 1 � B 1 ′ s 2 � B 2 ′ s 1 � B 3 s 3 � B 4 s 3 � B 5 B 0

Cluster Structures of Double Bott-Samelson Cells Cluster Structures We actually consider two versions of decorated double Bott-Samelson cells, one for G sc and one for G ad (analogues of the simply-connected form and the adjoint form in the semisimple cases).

Cluster Structures of Double Bott-Samelson Cells Cluster Structures We actually consider two versions of decorated double Bott-Samelson cells, one for G sc and one for G ad (analogues of the simply-connected form and the adjoint form in the semisimple cases). The natural projection G sc → G ad gives rise to natural projection maps A sc → A ad and p : Conf b d ( A sc ) → Conf b d ( A ad ).