Диссертация (1136188), страница 48

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Namely, we deduce a partial differential equation withconstant coefficients on the exponential generating function forthese numbers. For some particular classes of Gelfand–Zetlin polytopes, the number of vertices can be given by explicit formulas.© 2013 Elsevier Inc. All rights reserved.Keywords:Gelfand–Zetlin polytopesGenerating functionsf -Vector1. Introduction and statement of resultsGelfand–Zetlin polytopes play an important role in representation theory [2,7,8], symplectic geometry [1] and in algebraic geometry [3–5]. Let λ1 · · · λs be a non-decreasing finite sequenceof integers, i.e. an integer partition.

The corresponding Gelfand–Zetlin polytope is a convex polytopein Rs(s−1)2defined by an explicit set of linear inequalities depending on λi . It will be convenient tos(s−1)by pairs of integers (i , j ), where i runs from 1 to s − 1, andlabel the coordinates u i , j in R 2j runs from 1 to s − i.

The inequalities defining the Gelfand–Zetlin polytope can be visualized by thefollowing triangular table:λ1λ2u 1,1λ3u 1,2u 2,1...u 2 , s −2......λsu 1 , s −1......u s−2,1u s−2,2u s−1,1E-mail address: vtimorin@hse.ru (V. Timorin).0097-3165/$ – see front matter © 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jcta.2013.02.003(GZ)P. Gusev et al. / Journal of Combinatorial Theory, Series A 120 (2013) 960–969961where every triple of numbers a, b, c that appear in the table as vertices of the triangleabcare subject to the inequalities a c b.Gelfand–Zetlin polytopes parameterize irreducible finite-dimensional representations of GLn (C).Namely, if V λ is the simple GLn (C)-module of highest weight λ, then there is a Gelfand–Zetlin basis in V λ , whose elements are labeled by integer points in GZ (λ). In particular, the number of integerpoints in GZ (λ) is equal to the dimension of V λ .In this paper, we discuss generating functions for the number of vertices in Gelfand–Zetlin polytopes.

We will use the multiplicative notation for partitions, e.g. 1i 1 2i 2 3i 3 will denote the partitionconsisting of i 1 copies of 1, i 2 copies of 2, and i 3 copies of 3. Given a partition p, we write GZ ( p )for the corresponding Gelfand–Zetlin polytope, and V ( p ) for the number of vertices in GZ ( p ). ThusGZ (12 22 ) denotes the Gelfand–Zetlin polytope, for which s = 4, λ1 = λ2 = 1, and λ3 = λ4 = 2.

Notethat the partition 12 20 32 is the same as 12 32 . In particular, the polytope GZ (12 20 32 ) coincides withGZ (12 32 ) and is combinatorially equivalent to GZ (12 22 ).Fix a positive integer k, and consider all partitions of the form 1i 1 · · · kik , where a priori some ofthe powers i j may be zero. We let E k denote the exponential generating function for the numbersV (1i 1 · · · kik ), i.e. the formal power seriesE k ( z1 , . . . , zk ) =V 1i 1 · · · k i ki 1 ,...,ik 0 z1i 1i1!i···zkkik !.Our first result is a partial differential equation on the function E k :Theorem 1.1.

The formal power series E k satisfies the following partial differential equation with constantcoefficients: ∂∂∂∂k∂−++E k = 0.···∂ z1 · · · ∂ zk∂ z1 ∂ z2∂ zk−1 ∂ zkE.g. we have√E 2 ( z1 , z2 ) = e z1 +z2 I 0 (2 z1 z2 ),E 1 ( z1 ) = e z1 ,where I 0 is the modified Bessel function of the first kind with parameter 0. This function can bedefined e.g. by its power expansionI 0 (t ) =∞tn.n!2n =0It is also useful to consider ordinary generating functions for the numbers V (1i 1 · · · kik ):G k ( y 1 , . . .

, yk ) =iiV 1i 1 · · · kik y 11 · · · ykk .i 1 ,...,ik 0We will also deduce equations on G k . These will be difference equations rather than differential equations. For any power series f in the variables y 1 , . . . , yk , define the action of the divided differenceoperator i on f asi ( f ) =f − f | y i =0yi.Theorem 1.2. The ordinary generating function G k satisfies the following equation1 · · · k − (1 + 2 ) · · · (k−1 + k ) G k = 0.962P.

Gusev et al. / Journal of Combinatorial Theory, Series A 120 (2013) 960–969It is known that the ordinary generating functions G k can be obtained from exponential generating functions E k by the Laplace transform. Thus Theorem 1.2 can in principle be deduced fromTheorem 1.1 and the properties of the Laplace transform. However, we will give a direct proof.For k = 1, 2 and 3, the generating functions G k can be computed explicitly.

It is easy to see thatG 1 ( y1 ) =11 − y1,1G 2 ( y1 , y2 ) =1 − y1 − y2.We will prove the following theorem:Theorem 1.3. The function G 3 (x, y , z) is equal to2xz − y (1 − x − z) − y 1 − 2(x + z) + (x − z)22(1 − x − z)((x + y )( y + z) − y ).The numbers V k,,m = V (1k 2 3m ) can be alternatively expressed as coefficients of certain polynomials:Theorem 1.4. The number V k,,m for k > 0, > 0, m > 0 is equal to the coefficient of xk zm in the polynomial1 − xz 1 + xz(1 + x)k++m (1 + z)k++m − (x + z)k++m .Set s = k + + m. Note that, since the term (x + z)s is homogeneous of degree s, the number V k,,m ,where k, , m > 0, is also equal to the coefficient with xk zm in the power series(1 − xz)(1 + x)s (1 + z)s.1 + xzThis implies the following explicit formula for the numbers V k,,m (k, , m > 0): V k,,m =kssi+2.(−1)mk−im−isskNote that the sumki =1si =1 (−1) k−iism −ican be expressed as the value of the generalized hypergeo-metric function 3 F 2 , namely, it is equal tosk−1sF (1, 1 − k, 1 − m; 2 + + m, 2 + km−1 3 2+ ; −1).Remark.

The authors of paper [6] also consider vertices of Gelfand–Zetlin polytopes. However,Gelfand–Zetlin polytopes are understood in [6] in a different sense than in this paper and in otherpapers we cite. Namely, the authors impose additional restrictions on coordinates u i , j : the sum ofcoordinates in every row of table (GZ) should be equal to a given integer. The integer points in thissmaller polytope parameterize vectors with a given weight in the Gelfand–Zetlin basis of V λ . Themain result of [6] is an explicit parameterization of vertices. The corresponding result in our settingis obvious.

Thus there is no immediate connection between the methods and results from [6] andfrom this paper. On the other hand, there may be a possibility of combining both approaches in thesetting of [6].2. Recurrence relationsLet R be the polynomial ring in countably many variables x1 , x2 , x3 , . .

. . Define a linear operatorA : R → R by its action on monomials: every monomial m is mapped to k −1 k−1A (m) =( x i j + x i j +1 )xim,jj =1j =1P. Gusev et al. / Journal of Combinatorial Theory, Series A 120 (2013) 960–969963where i 1 < · · · < ik are the indices of all variables xi j that have positive exponents in m. Thus we haveby definition:A (1) = 1,A (x1 ) = 1,A (x1 x2 ) = x1 + x2 ,A (x1 x2 x3 ) = (x1 + x2 )(x2 + x3 ).The operator A thus defined reduces the degrees of all nonconstant polynomials. Therefore, for anypolynomial P , there exists a positive integer N such that A N ( P ) is a constant, which is independentof the choice of N provided that N is sufficiently large. We let A ∞ ( P ) denote this constant.Proposition 2.1. We have ii V 1i 1 · · · kik = A ∞ x11 · · · xkk .Proof. Some of the exponents i j may be zero.

The corresponding terms can be eliminated from boththe left-hand side and the right-hand side. We can then shift the remaining indices to reduce thestatement to its original form but with all exponents strictly positive. For example, the statementV (12 20 32 ) = A ∞ (x21 x02 x23 ) reduces to the statement V (12 32 ) = A ∞ (x21 x23 ) and then to the statementV (12 22 ) = A ∞ (x21 x22 ). Thus we may assume that all the exponents i j are strictly positive.We will argue by induction on the degree i 1 + · · · + ik , equivalently, on the dimension of theGelfand–Zetlin polytope GZ (1i 1 · · · kik ). Let π be the linear projection of GZ (1i 1 · · · kik ) to the cube Cgiven in coordinates (u 1 , .

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