Pieri Formula Research Articles

Pieri-Type Formulas for Nonsymmetric Macdonald Polynomials

In symmetric Macdonald polynomial theory, the Pieri formula gives the branching coefficients for the product of the rth elementary symmetric function e r (z) and the Macdonald polynomial $P_{\kappa}(z)$ . In this paper we give the nonsymmetric analogs for the cases r=1 and r= n−1. We do this by first deducing the decomposition for the product of any nonsymmetric Macdonald polynomial $E_{\eta}(z)$ with z i in terms of nonsymmetric Macdonald polynomials. As a corollary of finding the branching coefficients of $e_1(z)E_{\eta}(z)$ we evaluate the generalized binomial coefficients $(^{\eta}_v)$ associated with the nonsymmetric Macdonald polynomials for $|\eta|=|v|+1$ .

Difference Equations and Pieri Formulas for G 2 Type Macdonald Polynomials and Integrability

We present a second explicit difference equation and—by duality symmetry—a second explicit Pieri formula for the Macdonald polynomials associated with the exceptional root system of type G2, complementing the celebrated difference equation and Pieri formula due to Macdonald. As by-product, we find a complete set of integrals for the corresponding trigonometric Ruijsenaars–Schneider type particle system.

Schubert calculus and equivariant cohomology of grassmannians

We give a description of equivariant cohomology of grassmannians that places the theory into a general framework for cohomology theories of grassmannians. As a result we obtain a formalism for equivariant cohomology where the basic results of equivariant Schubert calculus, the basis theorem, Pieri's formula and Giambelli's formula can be obtained from the corresponding results of the general framework by a change of basis. In order to show that our formalism reflects the geometry of grassmannians we relate our theory to the treatment of equivariant cohomology of grassmannians by A. Knutson and T. Tao.

Macdonald Polynomials and Multivariable Basic Hypergeometric Series

We study Macdonald polynomials from a basic hypergeometric series point of view. In particular, we show that the Pieri formula for Macdonald polynomials and its recently discovered inverse, a recursion formula for Macdonald polynomials, both represent multivariable extensions of the terminating very-well-poised 6-phi-5 summation formula. We derive several new related identities including multivariate extensions of Jackson's very-well-poised 8-phi-7 summation. Motivated by our basic hypergeometric analysis, we propose an extension of Macdonald polynomials to Macdonald symmetric functions indexed by partitions with complex parts. These appear to possess nice properties.

Pieri’s formula for generalized Schur polynomials

Young's lattice, the lattice of all Young diagrams, has the Robinson-Schensted-Knuth correspondence, the correspondence between certain matrices and pairs of semi-standard Young tableaux with the same shape. Fomin introduced generalized Schur operators to generalize the Robinson-Schensted-Knuth correspondence. In this sense, generalized Schur operators are generalizations of semi-standard Young tableaux. We define a generalization of Schur polynomials as expansion coefficients of generalized Schur operators. We show that the commutating relation of generalized Schur operators implies Pieri's formula to generalized Schur polynomials.

A Pieri-type formula for the ${K}$-theory of a flag manifold

We derive explicit Pieri-type multiplication formulas in the Grothendieck ring of a flag variety. These expand the product of an arbitrary Schubert class and a special Schubert class in the basis of Schubert classes. These special Schubert classes are indexed by a cycle which has either the form (k-p+1, k-p+2,..., k+1) or the form (k+p, k+p-1,..., k), and are pulled back from a Grassmannian projection. Our formulas are in terms of certain labeled chains in the k-Bruhat order on the symmetric group and are combinatorial in that they involve no cancellations. We also show that the multiplicities in the Pieri formula are naturally certain binomial coefficients.

Equivariant Giambelli and Determinantal Restriction Formulas for the Grassmannian

The main result of the paper is a determinantal formula for the restriction to a torus fixed point of the equivariant class of a Schubert subva- riety in the torus equivariant integral cohomology ring of the Grassmannian. As a corollary, we obtain an equivariant version of the Giambelli formula. The (torus) equivariant cohomology rings of flag varieties in general and of the Grassmannian in particular have recently attracted much interest. Here we con- sider the equivariant integral cohomology ring of the Grassmannian. Just as the ordinary Schubert classes form a module basis over the ordinary cohomology ring of a point (namely the ring of integers) for the ordinary integral cohomology ring of the Grassmannian, so do the equivariant Schubert classes form a basis over the equivariant cohomology of a point (namely the ordinary cohomology ring of the classifying space of the torus) for the equivariant cohomology ring (this is true for any generalized flag variety of any type, not just the Grassmannian). Again as in the ordinary case, computing the structure constants of the multiplication with respect to this basis is an interesting problem that goes by the name of Schubert calculus. There is a forgetful functor from equivariant cohomology to ordinary cohomology so that results about the former specialize to those about the latter. Knutson-Tao-Woodward (5) and Knutson-Tao (6) show that the structure con- stants, both ordinary and equivariant, count solutions to certain jigsaw puzzles, thereby showing that they are manifestly positive. In the present paper we take a very different route to computing the equivariant structure constants. Namely, we try to extend to the equivariant case the classical approach by means of the Pieri and Giambelli formulas. Recall, from (3, Eq.(10), p.146) for example, that the Gi- ambelli formula expresses an arbitrary Schubert class as a polynomial with integral coefficients in certain Schubert classes—the Chern classes of the tautolog- ical quotient bundle—and that the Pieri formula expresses as a linear combination of the Schubert classes the product of a special Schubert class with an arbitrary Schubert class. Together they can be used to compute the structure constants. We only partially succeed in our attempt: the first of the three theorems of this paper—see §2 below—is an equivariant Giambelli formula that specializes to the ordinary Giambelli formula as in (3, Eq.(10), p.146), but we still do not have a satisfactory equivariant Pieri formula—see, however, §7 below. The derivation in Fulton (2, §14.3) of the Giambelli formula can perhaps be extended to the equi- variant case, but this is not what we do. Instead, we deduce the Giambelli formula from our second theorem which gives a certain closed-form determinantal formula for the restriction to a torus fixed point of an equivariant Schubert class.

Inversion of the Pieri formula for Macdonald polynomials

We give the explicit analytic development of Macdonald polynomials in terms of “modified complete” and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments for monomial, Jack and Hall–Littlewood symmetric functions.

Schubert Calculus via Hasse-Schmidt Derivations

A natural Hasse-Schmidt derivation on the exterior algebra of a free module realizes the (small quantum) cohomology ring of the grassmannian Gk(Cn) as a ring of operators on the exterior algebra of a free module of rank n. Classical Pieri's formula can be interpreted as Leibniz's rule enjoyed by special Schubert cycles with respect to the wedge product.

An analytic formula for Macdonald polynomials

We give the explicit analytic development of any Jack or Macdonald polynomial in terms of elementary (resp. modified complete) symmetric functions. These two developments are obtained by inverting the Pieri formula. To cite this article: M. Lassalle, M. Schlosser, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Littlewood–Richardson rules for Grassmannians

The classical Littlewood-Richardson rule (LR) describes the structure constants obtained when the cup product of two Schubert classes in the cohomology ring of a complex Grassmannian is written as a linear combination of Schubert classes. It also gives a rule for decomposing the tensor product of two irreducible polynomial representations of the general linear group into irreducibles, or equivalently, for expanding the product of two Schur S-functions in the basis of Schur S-functions. In this paper we give a short and self-contained argument which shows that this rule is a direct consequence of Pieri's formula (P) for the product of a Schubert class with a special Schubert class. There is an analogous Littlewood-Richardson rule for the Grassmannians which parametrize maximal isotropic subspaces of Cn, equipped with a symplectic or orthogonal form. The precise formulation of this rule is due to Stembridge (St), working in the context of Schur's Q-functions (S); the connection to geometry was shown by Hiller and Boe (HB) and Pragacz (Pr). The argument here for the type A rule works equally well in these more difficult cases and givesa simple derivation of Stembridge's rule from the Pieri formula of (HB). Currently there are many proofs available for the classical Littlewood-Richardson rule, some of them quite short. The proof of Remmel and Shimozono (RS) is also based on the Pieri rule; see the recent survey of van Leeuwen (vL) for alternatives. In contrast, we know of only two prior approaches to Stembridge's rule (described in (St, HH) and (Sh), respectively), both of which are rather involved. The argument presented here proceeds by defining an abelian group H with a basis of Schubert symbols, and a bilinear product on H with structure constants coming from the Littlewood-Richardson rule in each case. Since this rule is com- patible with the Pieri products, it suffices to show thatH is an associative algebra. The proof of associativity is based on Schutzenberger slides in type A, and uses the more general slides for marked shifted tableaux due to Worley (W) and Sagan (Sa) in the other Lie types. In each case, we need only basic properties of these operations which are easily verified from the definitions. Our paper is self-contained, once the Pieri rules are granted. The work on this article was completed during a fruitful visit to the Mathematisches Forschungsinstitut Oberwolfach, as part of the Research in Pairs program. It is a pleasure to thank the Institut for its hospitality and stimulating atmosphere.

Quantum Cohomology of Grassmannians

We give elementary proofs of the main theorems about the (small) quantum cohomology of Grassmannians, including the quantum Giambelli and quantum Pieri formulas, the rim-hook algorithm, the presentation, and a recent theorem of Fulton and Woodward about the minimal q-power which appears in a product of two Schubert classes.

A Pieri-Type Formula for H*T(SLn([formula omitted])/B)

The singular cohomology of the Grassmann variety of k-planes in Cn has a basis {sν} indexed by partitions. The classical Pieri formula is an explicit rule for determining the coefficients in the expansion of the cup products1m∪sλ=∑c1m,λμsμ,where 1m is a column of length m and s1m is the mth Chern class of the tautological bundle. Lascoux and Schutzenberger [C. R. Acad. Sci. Paris294 (1982), 447–450] formulated a generalization of Pieri's formula to the cohomology of the flag variety SLn(C)/B and briefly indicated an algebraic proof. (Manivel [Cours Spécialisés 3 (1998)] provides details of this proof.) A geometric proof was given by Sottile [Ann. Inst. Fourier (Grenoble) 46 (1996), 89–110]. In this paper we state and prove a generalization of this Pieri-type formula for the T-equivariant cohomology of the flag variety. We use the algebraic description of the T-equivariant cohomology of the flag variety due to Kostant and Kumar [Adv. Math.62 (1986), 187–237] and Arabia [Bull. Soc. Math. France117 (1989), 129–165], and our new formula exposes an equality of certain structure Constants in this algebra. Our proof is an induction based on the original idea in Lascoux and Schützenberger.

A proof of Pieri’s formula using the generalized Schensted insertion algorithm for rc-graphs

We provide a generalization of the Schensted insertion algorithm for rc-graphs of Bergeron and Billey. The new algorithm is used to give a new proof of Pieri's formula.

Numerical Schubert Calculus by the Pieri Homotopy Algorithm

Based on Pieri's formula on Schubert varieties, the Pieri homotopy algorithm was first proposed by Huber, Sottile, and Sturmfels [J. Symbolic Comput., 26 (1998), pp. 767--788] for numerical Schubert calculus to enumerate all p-planes in ${\mathbb C}^{m+p}$ that meet n given planes in general position. The algorithm has been improved by Huber and Verschelde [SIAM J. Control Optim., 38 (2000), pp. 1265--1287] to be more intuitive and more suitable for computer implementations. A different approach of employing the Pieri homotopy algorithm for numerical Schubert calculus is presented in this paper. A major advantage of our method is that the polynomial equations in the process are all square systems admitting the same number of equations and unknowns. Moreover, the degree of each polynomial equation is always 2, which warrants much better numerical stability when the solutions are being solved. Numerical results for a big variety of examples illustrate that a considerable advance in speed as well as much smaller storage requirements have been achieved by the resulting algorithm.

Skew Schubert functions and the Pieri formula for flag manifolds

We show the equivalence of the Pieri formula for flag manifolds and certain identities among the structure constants, giving new proofs of both the Pieri formula and of these identities. A key step is the association of a symmetric function to a finite poset with labeled Hasse diagram satisfying a symmetry condition. This gives a unified definition of skew Schur functions, Stanley symmetric function, and skew Schubert functions (defined here). We also use algebraic geometry to show the coefficient of a monomial in a Schubert polynomial counts certain chains in the Bruhat order, obtaining a new combinatorial construction of Schubert polynomials.

Semisymmetric polynomials and the invariant theory of matrix vector pairs

We introduce and investigate a one-parameter family of multivariate polynomials R λ R_\lambda . They form a basis of the space of semisymmetric polynomials, i.e., those polynomials which are symmetric in the variables with odd and even index separately. For two values of the parameter r r , namely r = 1 2 r=\frac 12 and r = 1 r=1 , the polynomials have a representation theoretic meaning related to matrix-vector pairs. In general, they form the semisymmetric analogue of (shifted) Jack polynomials. Our main result is that the R λ R_\lambda are joint eigenfunctions of certain difference operators. From this we deduce, among others, the Extra Vanishing Theorem, Triangularity, and Pieri Formulas.

Construction of commuting difference operators for multiplicity free spaces

The analysis of invariant differential operators on certain multiplicity free spaces led recently to the introduction of a family of symmetric polynomials that is more general than Jack polynomials (see [KS], but also [OO1], [OO2]). They are called interpolation Jack polynomials, shifted Jack polynomials, or Capelli polynomials. Apart from being inhomogeneous, they are distinguished from classical Jack polynomials by their very simple definition in terms of vanishing conditions. One of the most important and non-obvious properties of Capelli polynomials is that they are eigenfunctions of certain explicitly given difference (as opposed to differential) operators (see [KS]). This readily implies that their top homogeneous term is in fact a (classical) Jack polynomial. Other consequences include a binomial theorem, a Pieri formula, and much more. It is well-known that Jack polynomials are tied to root systems of type A and that they have natural analogues for other root systems (see e.g., [He]). Therefore, it is a natural problem as to whether this holds for the Capelli polynomials as well. Okounkov [Ok] proposed such an analogue for root systems of type BC and proved that these share some of the nice properties of Capelli polynomials. But, unfortunately, Okounkov’s polynomials do not satisfy difference equations. Also their representation-theoretic significance is not clear. In this paper we go back to the origin and let ourselves be guided by the theory of multiplicity free actions. It is known (see section 7 for details) that these actions give rise to combinatorial structures consisting of four data (Γ,Σ,W, l). Here Γ is a lattice, Σ ⊂ Γ a basis, W ⊆ AutΓ a finite reflection group, and l ∈ Γ some element. These data alone suffice to formulate the definition of a (generalized) Capelli polynomial but, in that generality, neither existence nor uniqueness will hold, let alone any other good property.

Schubert calculus and threshold polynomials of affine fusion

We show how the threshold level of affine fusion, the fusion of Wess-Zumino-Witten (WZW) conformal field theories, fits into the Schubert calculus introduced by Gepner. The Pieri rule can be modified in a simple way to include the threshold level, so that calculations may be done for all (non-negative integer) levels at once. With the usual Giambelli formula, the modified Pieri formula deforms the tensor product coefficients (and the fusion coefficients) into what we call threshold polynomials. We compare them with the q-deformed tensor product coefficients and fusion coefficients that are related to q-deformed weight multiplicities. We also discuss the meaning of the threshold level in the context of paths on graphs.

Combinatorial Aspects of the K-Theory of Grassmannians

In this paper we study Grothendieck polynomials indexed by Grassmannian permutations, which are representatives for the classes corresponding to the structure sheaves of Schubert varieties in the K-theory of Grassmannians. These Grothendieck polynomials are nonhomogeneous symmetric polynomials whose lowest homogeneous component is a Schur polynomial. Our treatment, which is closely related to the theory of Schur functions, gives new information about these polynomials. Our main results are concerned with the transition matrices between Grothendieck polynomials indexed by Grassmannian permutations and Schur polynomials on the one hand and a Pieri formula for these Grothendieck polynomials on the other.