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Here s1 and s2 denote the reflections associated with the shorter and longer simple roots, respectively. One can show thatthis polytope is defined by 8 inequalities and moreover one can choose coordinates(y1 , y2 , y3 , y4 ) in R4 so that precisely 4 inequalities become homogeneous, namely,0 ≤ y1 , 0 ≤ 2y4 ≤ y3 ≤ 2y2 [9, Example 3.4], [8, Example 2.9]. However, Pλ iscombinatorially different from the string and FFLV polytopes [10, Section 3.4]. Thefaces of Pλ that contain 0 can be encoded by the following diagram:+ ⇐⇒ 0 = y1+ ⇐⇒ 0 = y4+ ⇐⇒ y4 = y23 , e.g.

{y1 = 0, y3 = 2y2 } is encoded by +.+ ⇐⇒ y3 = 2y2+13Geometric mitosis reduces to a simple combinatorial rule. According to this rule theSchubert cycles on Sp4 /B can be represented by the following unions of faces of Pλ .+++ , Ss1 =Sid = {0} =+++∪ +S s1 s2 =+++ , Ss2 = + + ,+, Ss2 s1 =++ , Ss1 s2 s1 =∪ +, Ss2 s1 s2 s1 = Pλ =+,+Ss2 s1 s2 =+ ∪.+4.3. Newton–Okounkov polytopes of flag varieties. Fix the decompositionw0 = (s1 )(s2 s1 )(s3 s2 s1 ) . .

. (sn−1 . . . s1 ) of the longest element w0 ∈ Sn . Here si :=(i i + 1) is the i-th elementary transposition(simple reflection in the case of thenWeyl group Sn ). Denote by d := 2 the length of w0 .Fix a complete flag of subspaces F • := (F 1 ⊂ F 2 ⊂ . . . ⊂ F n−1 ⊂ Cn ) (thisamounts to fixing a Borel subgroup B ⊂ GLn ). Also fix a basis e1 ,.

. . , en in Cncompatible with F • (or a maximal torus in B), that is, F i = he1 , . . . , ei i. In whatfollows, w` for ` = 1,. . . , d denotes the subword of w0 obtained by deleting the first `simple reflections in w0 , and w` denotes the corresponding element of Sn . Considerthe flag of translated Schubert subvarieties:−1−1w0 Xid ⊂ w0 wd−1Xwd−1 ⊂ w0 wd−2Xwd−2 ⊂ . . . ⊂ w0 w1−1 Xw1 ⊂ GLn /B,(∗)where Schubert subvarieties are taken with respect to the flag F • , i.e., Xw =BwB/B. Recall that the open Schubert cell C with respect to F • is defined asthe set of all flags M • that are in general position with the standard flag F • ,i.e., all intersections M i ∩ F j are transverse. Let y1 , .

. . , yd be coordinates onthe open Schubert cell C (with respect to F • ) that are compatible with (∗), i.e.,w0 w`−1 Xw` ∩ C = {y1 = . . . = y` = 0}.For instance, we can identify the open Schubert cell C with an affine space Cd bychoosing for every flag M • a basis v1 ,. . . , vn in Cn of the form:v1 = en + xn−1en−1 + . . . + x11 e1 ,1v2 = en−1 + xn−2en−2 + . . . + x12 e1 ,2..., vn−1 = e2 + x1n−1 e1 ,vn = en ,so that M i = hv1 , . . .

, vi i. Such a basis is unique, hence, the coefficients (xij )i+j<nare coordinates on the open cell. It is not hard to check that the coordinates(y1 , . . . , yd ) := (x1n−1 ; x1n−2 , x2n−2 ; . . . ; x11 , x21 , . . . , x1n−1 ) are compatible with (∗). In14other words, every flag M • ∈ C gets identified with a triangular matrix: 1x1 x12 . . . x1n−1 1 x21 x22 . . .10 .., ....0xn−1 1 . . .0011000and we order the coefficients (xij )i+j<n of this matrix by starting from column (n−1)and going from top to bottom in every column and from right to left along columns.In [10, Section 2.2] we give another example of coordinates compatible with (∗).The latter coordinates are more natural from the geometric viewpoint and are relatedto geometry of the Bott–Samelson variety associated with w0 .Fix the lexicographic ordering on monomials in coordinates y1 , .

. . , yd so thaty1  y2  . . .  yd . Let v denote the lowest order term valuation on C(Xw0 ) =C(GLn /B) associated with these coordinates and ordering. Let Lλ be the linebundle on GLn /B corresponding to a dominant weight λ := (λ1 , . . . , λn ) ∈ Znof GLn . Denote by ∆v (GLn /B, Lλ ) ⊂ Rd the Newton–Okounkov convex bodycorresponding to GLn /B, Lλ and v.Theorem 4.4. [10, Theorem 2.1] The Newton–Okounkov convex body∆v (GLn /B, Lλ ) coincides with the Feigin–Fourier–Littelmann–Vinberg polytopeF F LV (λ).We now recall the definition of F F LV (λ). Label coordinates in Rd correspondingto (y1 , .

. . , yd ) by (u1n−1 ; u2n−2 , u1n−2 ; . . . ; un−1, un−2, . . . , u11 ). Arrange the coordinates11into the tableλ1λ2λ3...λn...u1n−1u12u11u21...u2n−2(F F LV )......un−21u1n−1u2n−2The polytope F F LV (λ) is defined by inequalities ulm ≥ 0 andXulm ≤ λi − λj(l,m)∈Dfor all Dyck paths going from λi to λj in table (F F LV ) where 1 ≤ i < j ≤ n.For instance, the computation of the polytope ∆v (GLn /B, Lλ ) for n = 3 andλ = (1, 0, −1) is illustrated in Examples 1.4, 1.7, 2.3.References[ACK] B. H. An, Yu. Cho, J.

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Math. Soc. Transl. Ser. 2, 181 (1998), 231–244E-mail address: vkiritch@hse.ruLaboratory of Algebraic Geometry and Faculty of Mathematics, National Research University Higher School of Economics, Russian Federation, Usacheva str.6, 119048 Moscow, Russia16Institute for Information Transmission Problems, Moscow, Russia.

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