Rhombic tilings and Bott-Samelson varieties
Laura Escobar, Oliver Pechenik, Bridget Eileen Tenner, Alexander Yong
RRHOMBIC TILINGS AND BOTT-SAMELSON VARIETIES
LAURA ESCOBAR, OLIVER PECHENIK, BRIDGET EILEEN TENNER, AND ALEXANDER YONGA
BSTRACT . S. Elnitsky (1997) gave an elegant bijection between rhombic tilings of n -gonsand commutation classes of reduced words in the symmetric group on n letters. P. Mag-yar (1998) found an important construction of the Bott-Samelson varieties introduced byH.C. Hansen (1973) and M. Demazure (1974). We explain a natural connection betweenS. Elnitsky’s and P. Magyar’s results. This suggests using tilings to encapsulate Bott-Samelson data (in type A ). It also indicates a geometric perspective on S. Elnitsky’s combi-natorics. We also extend this construction by assigning desingularizations to the zonotopaltilings considered by B. Tenner (2006).
1. I
NTRODUCTION
Let X = Flags ( C n ) be the variety of complete flags C ⊂ F ⊂ F ⊂ · · · ⊂ F n − ⊂ C n . The group GL n ( C ) acts on the variety X by change of basis, as does its subgroup B of invertible upper triangular matrices and its maximal torus T of invertible diagonalmatrices. The T -fixed points are in bijection with permutations w in the symmetric group S n : they are the flags F ( w ) • defined by F ( w ) k = (cid:104) (cid:126)e w (1) , (cid:126)e w (2) , . . . , (cid:126)e w ( k ) (cid:105) where (cid:126)e i is the i -thstandard basis vector. The Schubert variety X w is the B -orbit closure of F ( w ) • .There is longstanding interest in singularities of Schubert varieties; see, for example,the text by S. Billey-V. Lakshmibai [BL00]. Famously, H.C. Hansen [Han73] and M. De-mazure [Dem74] independently presented (in all Lie types) resolutions of singularities BS ( i ,i ,...,i (cid:96) ( w ) ) of X w , one for each reduced word s i s i · · · s i (cid:96) ( w ) of w . M. Demazure calledthese resolutions Bott-Samelson varieties in reference to a related construction of R. Bott-H. Samelson [BS55]. In more recent work, P. Magyar [Mag98] found an important descrip-tion of Bott-Samelson varieties.We propose a canonical connection between P. Magyar’s work and the rhombic tilingsof S. Elnitsky [Eln97]. In this way, tilings graphically encapsulate Bott-Samelson data.(One should compare what follows to the similar use of X. Viennot’s heaps [Vie89] topresent Bott-Samelsons; see N. Perrin’s [Per07] and B. Jones-A. Woo’s [JW13].)Given a permutation w ∈ S n , the Elnitsky n -gon E ( w ) has sides of length one, andthese are labeled, in order, by , , . . . , n, w ( n ) , w ( n − , . . . , w (1) , in which the first n labelsform half of a regular n -gon, and sides with the same label are parallel. In Figure 1, wegive the Elnitsky -gon for the permutation ∈ S ; this example will be referencedthroughout this work.Let T ( w ) be the set of rhombic tilings of E ( w ) in which the rhombi have sides of lengthone and edges parallel to edges of E ( w ) . The main result of S. Elnitsky’s aforementioned Date : June 9, 2016. a r X i v : . [ m a t h . C O ] J un C C C C C C C G G G G G G F IGURE
1. The rhombic tiling picture of Bott-Samelson varieties, for thepolygon E (7456312) .work is that the set T ( w ) is in bijection with the commutation classes of reduced wordsof w [Eln97, Theorem 2.2].We associate a vector space to each vertex of a tiling T ∈ T ( w ) . Starting with the vertexbetween the edges labeled and w (1) , label the vertices of E ( w ) in clockwise order by C , C , . . . , C n , G n − , G n − , . . . , G . In general, let V x be the vector space associated to a vertex x in the tiling. The dimensionof V x is the minimal path length from C to x along tile edges. In Figure 1, we have onlylabeled the external vertices.For adjacent vertices x and y in a tiling T ∈ T ( w ) , write x → y if dim( V x ) + 1 = dim( V y ) .Let Vert ( T ) be the vertices of T , and define Z T := (cid:8) ( V x : x ∈ Vert ( T )) : V y ⊆ V z if y → z (cid:9) ⊂ (cid:89) x ∈ Vert ( T ) Gr dim( V x ) ( C n ) , where Gr k ( C n ) is the Grassmannian of k -dimensional subspaces of C n .Define the map π : Z T → X by forgetting all vector spaces except those labeled by thevertices G , G , . . . , G n − . In our example, π maps the point depicted in Figure 1 to thecomplete flag C ⊂ G ⊂ G ⊂ G ⊂ G ⊂ G ⊂ G ⊂ C . The following theorem suggests a Schubert-geometric interpretation of tilings of Elnit-sky polygons.
Theorem 1.1.
For T ∈ T ( w ) , Z T is a Bott-Samelson variety, i.e., a desingularization π : Z T → X w . Conversely, every Bott-Samelson variety BS ( i ,...,i (cid:96) ( w ) ) is canonically isomorphic to Z T forsome T ∈ T ( w ) where w = s i . . . s i (cid:96) ( w ) and T is given in an explicit manner by [Eln97, Theo-rem 2.2] . In Section 2, we prove Theorem 1.1. The remainder of this paper concerns other Bott-Samelson data encoded by tilings. In Section 3, we explain how the hexagon flips of Eln97, Section 3] may be interpreted geometrically. This naturally leads to the zonotopaltilings of [Ten06], each of which corresponds to a desingularization of a Schubert variety.We collect some additional discussion in Section 4; in particular, we explain how coloringrhombi of a tiling describes T -fixed points as well as a standard stratification of a Bott-Samelson variety. 2. P ROOF OF T HEOREM w are commutation equivalent if they can be obtained fromone another using only the relation s i s j = s j s i when | i − j | > .By [Eln97, Theorem 2.2], the set T ( w ) bijects with commutation classes of reducedwords of w . (Note that our orientation of the polygon is a horizontal reflection of theorientation given in [Eln97].) To link with [Mag98], we recall the bijection. Consider atiling T ∈ T ( w ) . The edges of T that coincide with edges of E ( w ) inherit the labels ofthose edges, and we label the interior edges of T so that parallel edges have the samelabels.Let B be the base boundary of E ( w ) , formed by the edges of the polygon appearingclockwise between C and C n . Pick any rhombus R of T that shares two edges with B . Set i := d + 1 , where d is the minimum distance from C to R . Remove R anddefine a new boundary, B , from B by using the other two edges of R instead. Nowrepeat this process: pick any rhombus R that shares two edges with B ; set i := d + 1 ,where d is the minimum distance from C to R ; remove R and form a new boundary B . Iterating this process an additional (cid:96) ( w ) − times produces ( i , i , . . . , i (cid:96) ( w ) ) , for which s i s i · · · s i (cid:96) ( w ) represents a commutation class of reduced words for w . The other directionof the bijection is indicated below.We now show that Z T is isomorphic to BS ( i ,i ,...,i (cid:96) ( w ) ) . P. Magyar [Mag98, Theorem 1]describes BS ( i ,i ,...,i (cid:96) ( w ) ) as a list ( F • , . . . , F m • ) of m + 1 flags where F • is the base flag, andsuch that F k • agrees with F k − • everywhere except possibly on the i k -th subspace. Such alist of flags transparently corresponds in a one-to-one fashion to points in Z T : F • is thebase flag which is on the base boundary B and in general, F k • is the flag on B k .Suppose that j = ( j , j , . . . ) is commutation equivalent to i = ( i , i , . . . ) . It is well-known to experts that BS i and BS j are isomorphic varieties, but we include a proof forcompleteness. It suffices to prove this when j = ( i , . . . , i k +1 , i k , . . . , i (cid:96) ( w ) ) differs from i only in positions k and k + 1 . The general result then follows by induction. Now, ( F • , . . . , F m • ) is equivalent to a list of subspaces ( V , V , . . . ) satisfying: • dim( V k ) = i k ; • C i − ⊂ V ⊂ C i +1 ; that is, V is contained in the ( i + 1) -dimensional subspace of F • and contains the ( i − -dimensional subspace of F • ; • V is contained in the ( i + 1) -dimensional subspace of F • and contains the ( i − -dimensional subspace of F • ; and so on.Since | i k +1 − i k | > , the ( i k + 1) -, ( i k − -, ( i k +1 + 1) -, and ( i k +1 − -dimensional subspacesof F k • are precisely the subspaces of F k − • with those dimensions. So if a generic elementof BS i is ( V , V , . . . ) , then a generic element of BS j is ( V , V , . . . , V k +1 , V k , . . . ) . That is, theisomorphism by switching factors: τ k : Gr i ( C n ) × · · · × Gr i k ( C n ) × Gr i k +1 ( C n ) × · · · → Gr i ( C n ) × · · · × Gr i k +1 ( C n ) × Gr i k ( C n ) × · · · estricts to a canonical isomorphism from BS ( i ,i ,... ) to BS ( i ,...,i k +1 ,i k ,... ) . In other words, T ( w ) indexes Bott-Samelson varieties up to commutation equivalence.Given i = ( i , i , . . . ) representing a commutation class for w (that is, s i s i · · · is a re-duced decomposition of w ), the inverse map to S. Elnitsky’s bijection constructs an or-dered tiling of E ( w ) , as follows. For k ≥ , set w ( k ) := s i s i · · · s i k . By [Eln97], for ≤ k ≤ (cid:96) ( w ) , the values w ( k ) ( i k ) and w ( k ) ( i k + 1) label adjacent edges of the bound-ary B k − . Place a rhombus, R k , so that two of its edges coincide with the edges labeled w ( k ) ( i k ) and w ( k ) ( i k + 1) in B k − , and define the new boundary B k from B k − by using theother two edges of R k . This explicitly picks Z T from i such that Z T ∼ = BS i , as desired. (cid:3) Example . Consider the tiling T ∈ T (7456312) depicted in Figure 1. One way to selectthe rhombi { R , R , . . . } described in the proof of Theorem 1.1 is shown in Figure 2, wherewe have recorded only the subscript k of the rhombus R k . The labeling in this figurerepresents the commutation class of the reduced word s s s s s s s s s s s s s s s s s for the permutation . Any other such labeling of these tiles would produce a dif-ferent, but commutation equivalent, reduced word. For example, the labeling obtained byswapping the selections for R and R , both of which share two edges with the bound-ary B , as indicated in Figure 2, produces the commutation equivalent reduced word s s s s s s s s s s s s s s s s s . F IGURE
2. A labeling of the rhombi in an element of T (7456312) , corre-sponding to the reduced word s s s s s s s s s s s s s s s s s for the per-mutation . The boundary B is indicated by thick line segments.3. F LIPS AND ZONOTOPAL TILINGS
Flips.
Any pair of rhombic tilings of E ( w ) are connected by a sequence of hexagon“flips” [Eln97, Section 3]. The effect of a single flip is depicted in Figure 3.This flip has a geometric interpretation. Let T, T (cid:48) ∈ T ( w ) be two rhombic tilings thatdiffer by a single flip. Let T H be the tiling of E ( w ) obtained from T (or, equivalently, from → F IGURE
3. Two elements of T (7456312) , related by a hexagon flip. T (cid:48) ) by erasing the three internal edges by which T and T (cid:48) differ, and placing a hexago-nal tile in the flip location. As before, associate vector spaces V x to each vertex x in T H ,where dim( V x ) equals the distance from x to C . The resulting space Z T H is similar to aBott-Samelson variety: instead of being (cid:96) ( w ) -fold iterated CP -bundles over the base flag,we replace three of these CP -bundles (corresponding to either triple of rhombi in thehexagon) by a Flags ( C ) -bundle. We then have Z T Z T Z T H f T f T where the two maps are the projections determined by forgetting the vector space at-tached to the internal vertex of the hexagon.3.2. Zonotopal tilings.
The tiling T H described above is a special case of the “zono-topal” tilings of Elnitsky polygons, which were studied by the third author in [Ten06].To be precise, a -zonotope is the projection of a regular q -dimensional cube onto the ( -dimensional) plane; equivalently, a -zonotope is a centrally symmetric convex polygon.A zonotopal tiling of a region is a tiling by -zonotopes. Figure 4 shows a zonotopal tilingof E (87465312) using one octagon, three hexagons, and ten rhombi.Let T zono ( w ) be the collection of zonotopal tilings of E ( w ) , in which the tiles ( -zonotopes)have sides of length one and edges parallel to edges of E ( w ) . Because rhombi are a typeof -zonotope, we have T ( w ) ⊆ T zono ( w ) .Given a zonotopal tiling Z ∈ T zono ( w ) , we can define its corresponding generalizedBott-Samelson variety Z Z by extending the construction from Section 3.1. For each vertex x in the zonotopal tiling, associate a vector space V x whose dimension is the minimal pathlength from the bottom vertex to x along tile edges. Define Z Z := { ( V x : x ∈ Vert ( Z )) : V y ⊆ V z if y → z } . Let T be a rhombic tiling that refines Z ; Z T may be constructed as iterated CP -bundlesover a point. In the analogous construction of Z Z , for each k -gon of Z , we replace k CP -bundles with a Flags ( C k ) -bundle. The variety Z Z is smooth of dimension (cid:96) ( w ) . De-fine π Z : Z Z → X w by forgetting all vector spaces except those labeled by the vertices G , G , . . . , G n − . C C C C C C C C G G G G G G G F IGURE
4. A zonotopal tiling for the permutation 87465312.
Theorem 3.1.
Given a zonotopal tiling Z ∈ T zono ( w ) , its corresponding generalized Bott-Samelsonvariety Z Z together with the map π Z : Z Z → X w is a resolution of singularities.Proof. Let π T : Z T → X w be a Bott-Samelson resolution where T is any rhombic tilingthat refines Z . We know that π T is birational, so let π (cid:48) T be its rational inverse. Let f : Z T (cid:16) Z Z be the projection determined by forgetting the vector spaces attached to theinternal vertices of the k -gons. Since f is surjective, the image of π Z is indeed X w andthe following commutative diagram implies that f ◦ π (cid:48) T is a rational inverse to π Z . Z T Z Z X wfπ T π Z It follows that π Z : Z Z → X w is also a resolution of singularities. (cid:3) The zonotopal tilings T zono ( w ) of E ( w ) have a natural poset structure, as studied by thethird author in [Ten06]. The order relation in this poset is given by reverse edge inclusion.Thus the rhombic tilings are the minimal elements in the poset. A pair of rhombic tilingsdiffer by a single hexagon flip if and only if they are covered by a common element.Similarly, one can get a broader sense of how closely two rhombic tilings (equivalently,two commutation classes of reduced words for w ) are related by determining their leastupper bound in this poset. Geometrically, the relations in the poset T zono ( w ) correspondto the projections Z Z (cid:16) Z Z (cid:48) between two generalized Bott-Samelsons for X w .By [Ten06, Theorem 6.13], the poset of zonotopal tilings of E ( w ) has a unique maximalelement ˆ Z exactly in the case that w avoids the patterns , , and . In thiscase, there is a distinguished Z ˆ Z with a projection Z Z (cid:16) Z ˆ Z from every other generalizedBott-Samelson. Such permutations have been enumerated by T. Mansour [Man06].For comparison, consider Elnitsky polygons whose zonotopal tilings do not containany hexagonal tiles (equivalently, those polygons with a unique zonotopal tiling). These orrespond to -avoiding permutations, which are exactly those whose reduced wordscontain no long braid moves [BJS93, Theorem 2.1] (see also [Ten15, Section 3] for moregeneral results relating pattern avoidance and reduced words). The unique tiling in thiscase is a deformation of the skew shape associated to the permutation by consideringits Rothe diagram and removing empty rows and columns. A standard filling orders thetilings in the sense of [Eln97] (and the final paragraph of the proof of Theorem 1.1).We now have the following result (cf. [Ele15, Remark 3.1], where this fact for ordinaryBott-Samelsons is noted).
Proposition 3.2.
Suppose that Z ∈ T zono ( w ) , and that the number of i -sided tiles in Z is t i , foreach i ≥ . Then the Poincar´e polynomial of the cohomology ring H (cid:63) ( Z Z ) is (cid:96) ( w ) (cid:88) k =0 dim H k ( Z Z ) q k = (cid:89) i ≥ [ i ] q ! t i , where [ i ] q := 1 + q + q + · · · + q i − and [ i ] q ! := [ i ] q [ i − q · · · · [1] q .Proof. The variety Z Z is constructed as iterated flag bundles over a point, where t i of the fi-brations are by Flags ( C i ) . It is a standard fact (following from the Schubert decompositionof Flags ( C i ) ) that the Poincar´e polynomial of H (cid:63) ( Flags ( C i )) is [ i ] q ! (indeed, [ i ] q ! is the ordi-nary generating function for S i with each permutation weighted by Coxeter length). Theproposition now follows from the Leray-Hirsch theorem (cf. [Hat02, Theorem 4D.1]). (cid:3)
4. A
DDITIONAL DISCUSSION C C C C C C C C F ( s )3 C F ( s s )4 F ( s )3 C C F ( s s s )5 F ( s s )4 F ( s )3 F ( s s )2 C F ( s s )2 F ( s )3 C C C F ( s )5 F IGURE
5. A coloring corresponding to a fixed point of Z T .One may reformulate certain results about BS i in terms of rhombic colorings; we referto [Esc16, Section 3.2] for background with further references. roposition 4.1. For T ∈ T ( w ) , the T -fixed points of Z T (under the diagonal action) are inone-to-one correspondence with bipartitions of the rhombi of T .Proof. Consider a -coloring of the rhombi of T representing the bipartition (as shown inFigure 5). There is a unique way to choose { V x } x ∈ Vert ( T ) such that(1) each V x is the span of a subset of the standard basis; and,(2) for any rhombus, its two vector spaces of common dimension are the same (resp.,different) if the rhombus is light-colored (resp., dark-colored).Since the T -action is diagonal, if { V x } x ∈ Vert ( T ) is a T -fixed point of Z T , then each V x must be T -fixed, i.e., each V x must be spanned by a subset of the standard basis { e , . . . , e n } . Usingthe required containment relations, we can inductively determine V x for each vertex of T by following an ordering of the rhombi given by a representative of the commutationclass of T . At a particular colored rhombus, we make the two vector spaces of commondimension the same (resp., different) if the rhombus is light-colored (resp., dark-colored). V c = V a (cid:76) (cid:104) e b , e c (cid:105) = (resp., (cid:54) = ) V b = V a (cid:76) (cid:104) e b (cid:105) V x V a Conversely, every T -fixed point can be indicated by such a coloring. (cid:3) M. Demazure [Dem74] used the T -fixed points to prove that the image of BS ( i ,i ,... ) under the Bott-Samelson map π is indeed the Schubert variety X s i s i ... . These fixed pointsare also useful in the study of moment polytopes of Bott-Samelson varieties, and for otherapplications.These colorings also correspond to a stratification of Z T by smaller Bott-Samelsons.Given a coloring, the corresponding stratum has the property that, for any light-coloredrhombus, its two vector spaces of common dimension are equal. The dark-colored rhombiimpose no conditions. The unique smallest stratum corresponds to the all-light coloring,whereas the unique largest stratum corresponds to the all-dark one. For background, see[Esc16, Section 4.3]In [Eln97], the author extends his main construction to the other Weyl groups of classi-cal Lie type. This seems related to the Bott-Samelsons for the associated Lie groups.A CKNOWLEDGEMENTS
We thank Allen Knutson and Alexander Woo for helpful comments. OP was supportedby an NSF Graduate Research Fellowship. BT was partially supported by a Simons Foun-dation Collaboration Grant for Mathematicians. AY was supported by an NSF grant.R
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EPARTMENT OF M ATHEMATICS , U. I
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