Pieri Rule Research Articles

Quasisymmetric Schur functions

We introduce a new basis for the algebra of quasisymmetric functions that naturally partitions Schur functions, called quasisymmetric Schur functions. We describe their expansion in terms of fundamental quasisymmetric functions and determine when a quasisymmetric Schur function is equal to a fundamental quasisymmetric function. We conclude by describing a Pieri rule for quasisymmetric Schur functions that naturally generalizes the Pieri rule for Schur functions. Nous étudions une nouvelle base des fonctions quasisymétriques, les fonctions de quasiSchur. Ces fonctions sont obtenues en spécialisant les fonctions de Macdonald dissymétrique. Nous décrivons les compositions que donne une simple fonction quasisymétriques. Nous décrivons aussi une règle par certaines fonctions de Schur.

Fusion algebras, symmetric polynomials, and [formula omitted]-orbits of [formula omitted

A method of computing fusion coefficients for Lie algebras of type A N − 1 on level k was recently developed by A. Feingold and M. Weiner [A. Feingold, M. Weiner, Type A fusion rules from elementary group theory, in: S. Berman, P. Fendley, Y.-Z. Huang, K. Misra, B. Parshall (Eds.), Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory, Proceedings of an International Conference on Infinite-Dimensional Lie Theory and Conformal Field Theory, May 23–27, 2000, University of Virginia, Charlottesville, Virginia, in: Contemp. Math., vol. 297, Amer. Math. Soc., Providence, RI, 2002, pp. 97–115] using orbits of Z N k under the permutation action of S k on k-tuples. They got the fusion coefficients only for N = 2 and 3. We will extend this method to all N ⩾ 2 and all k ⩾ 1 . First we show a connection between Young diagrams and S k -orbits of Z N k , and using Pieri rules we prove that this method works for certain specific weights that generate the fusion algebra. Then we show that the orbit method does not work in general, but with the help of the Jacobi–Trudi determinant, we give an iterative method to reproduce all type A fusion products.

SUPER RSK-ALGORITHMS AND SUPER PLACTIC MONOID

We construct the analog of the plactic monoid for the super semistandard Young tableaux over a signed alphabet. This is done by developing a generalization of the Knuth's relations. Moreover we get generalizations of Greene's invariants and Young–Pieri rule. A generalization of the symmetry theorem in the signed case is also obtained. Except for this last result, all the other results are proved without restrictions on the orderings of the alphabets.

Schur Functions and Alternating Sums

We discuss several well known results about Schur functions that can be proved using cancellations in alternating summations; notably we shall discuss the Pieri and Murnaghan-Nakayama rules, the Jacobi-Trudi identity and its dual (Von Nägelsbach-Kostka) identity, their proofs using the correspondence with lattice paths of Gessel and Viennot, and finally the Littlewood-Richardson rule. Our our goal is to show that the mentioned statements are closely related, and can be proved using variations of the same basic technique. We also want to emphasise the central part that is played by matrices over $\{0,1\}$ and over $\Bbb N$; we show that the Littlewood-Richardson rule as generalised by Zelevinsky has elegant formulations using either type of matrix, and that in both cases it can be obtained by two successive reductions from a large signed enumeration of such matrices, where the sign depends only on the row and column sums of the matrix.

Equivariant quantum Schubert calculus

We study the T -equivariant quantum cohomology of the Grassmannian. We prove the vanishing of a certain class of equivariant quantum Littlewood–Richardson coefficients, which implies an equivariant quantum Pieri rule. As in the equivariant case, this implies an algorithm to compute the equivariant quantum Littlewood–Richardson coefficients.

Tableaux on [formula omitted]-cores, reduced words for affine permutations, and k-Schur expansions

The k -Young lattice Y k is a partial order on partitions with no part larger than k . This weak subposet of the Young lattice originated (Duke Math. J. 116 (2003) 103–146) from the study of the k -Schur functions s λ ( k ) , symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by k -bounded partitions. The chains in the k -Young lattice are induced by a Pieri-type rule experimentally satisfied by the k -Schur functions. Here, using a natural bijection between k -bounded partitions and k + 1 -cores, we establish an algorithm for identifying chains in the k -Young lattice with certain tableaux on k + 1 cores. This algorithm reveals that the k -Young lattice is isomorphic to the weak order on the quotient of the affine symmetric group S ˜ k + 1 by a maximal parabolic subgroup. From this, the conjectured k -Pieri rule implies that the k -Kostka matrix connecting the homogeneous basis { h λ } λ ∈ Y k to { s λ ( k ) } λ ∈ Y k may now be obtained by counting appropriate classes of tableaux on k + 1 -cores. This suggests that the conjecturally positive k -Schur expansion coefficients for Macdonald polynomials (reducing to q , t -Kostka polynomials for large k ) could be described by a q , t -statistic on these tableaux, or equivalently on reduced words for affine permutations.

A Positive Proof of the Littlewood-Richardson Rule using the Octahedron Recurrence

We define the hive ring, which has a basis indexed by dominant weights for $GL_n({\Bbb C})$, and structure constants given by counting hives [Knutson-Tao, "The honeycomb model of $GL_n$ tensor products"] (or equivalently honeycombs, or BZ patterns [Berenstein-Zelevinsky, "Involutions on Gel$'$fand-Tsetlin schemes$\dots$ "]). We use the octahedron rule from [Robbins-Rumsey, "Determinants$\dots$"] to prove bijectively that this "ring" is indeed associative. This, and the Pieri rule, give a self-contained proof that the hive ring is isomorphic as a ring-with-basis to the representation ring of $GL_n({\Bbb C})$. In the honeycomb interpretation, the octahedron rule becomes "scattering" of the honeycombs. This recovers some of the "crosses and wrenches" diagrams from Speyer's very recent preprint ["Perfect matchings$\dots$"], whose results we use to give a closed form for the associativity bijection.

A Pieri rule for Hermitian symmetric pairs I

Let X be an irreducible Hermitian symmetric space of non-compact type and rank r. Let p ∈ X and let K be the isotropy group of p in the group of biholomorphic transformations. Let S denote the symmetric algebra in the holomorphic tangent space to X at p. Then S is multiplicity free as a representation of K and the irreducible constituents are parametrized by r-tuples, (m 1 ...,m r ) with m 1 >... > m r > 0. That is, the same parameters as the irreducible polynomial representations of GL(r). Let S[m 1 ,...,m r ] be the corresponding isotypic component. In this article we show that the product in S, S[m 1 ,...,m r ]S[k,0,0,...,0] is a direct sum of constituents following precisely the classical Pieri rule.

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.

Projective resolutions of q-Weyl modules

Here projective resolutions for the q-Weyl modules over the q-Schur algebra are given using the quantum multilinear algebra of Hashimoto and Hayashi and the implicit description of projective modules over the q-Schur algebra as tensor products of quantum divided powers. They are given by a mapping cone construction on short exact sequences of generalized Weyl modules attached to shapes and are a specific instance of a general algorithm included in the paper. The proof of the short exactness of these sequences requires a proof of the Pieri rule, giving a filtration for the tensor product of a q-Weyl module with a quantum tensor divided power with sections that are q-Weyl modules, here proven for the quantum situation. The proof further builds on the letterplace techniques developed by Rota and Buchsbaum and realized for the quantum situation as codeterminants by Green.

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.

Quantum double Schubert polynomials, quantum Schubert polynomials and Vafa–Intriligator formula

We study algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quantum double Schubert polynomials S ̃ w(x,y) , which are the Lascoux–Schützenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy identity. We define also quantum Schubert polynomials S ̃ w(x) as the Gram–Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Using quantum Cauchy identity, we prove that S ̃ w(x)= S ̃ w(x,y)| y=0 and as a corollary obtain a simple formula for the quantum Schubert polynomials S ̃ w(x)=∂ ww 0 (y) S ̃ w 0 (x,y)| y=0 . We also prove the higher genus analog of Vafa–Intriligator's formula for the flag manifolds and study the quantum residues generating function. We introduce the Ehresmann–Bruhat graph on the symmetric group and formulate the equivariant quantum Pieri rule.

Lattice Diagram Polynomials and Extended Pieri Rules

The lattice cell in thei+1 st row andj+1 st column of the positive quadrant of the plane is denoted (i,j). Ifμis a partition ofn+1, we denote byμ/ijthe diagram obtained by removing the cell (i,j) from the (French) Ferrers diagram ofμ. We setΔμ/ij=det‖xpjiyqji‖ni,j=1, where (p1,q1),…,(pn,qn) are the cells ofμ/ij, and letMμ/ijbe the linear span of the partial derivatives ofΔμ/ij. The bihomogeneity ofΔμ/ijand its alternating nature under the diagonal action ofSngivesMμ/ijthe structure of a bigradedSn-module. We conjecture thatMμ/ijis always a direct sum ofkleft regular representations ofSn, wherekis the number of cells that are weakly north and east of (i,j) inμ. We also make a number of conjectures describing the precise nature of the bivariate Frobenius characteristic ofMμ/ijin terms of the theory of Macdonald polynomials. On the validity of these conjectures, we derive a number of surprising identities. In particular, we obtain a representation theoretical interpretation of the coefficients appearing in some Macdonald Pieri Rules.

BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials

We consider 3-parametric polynomialsP *μ (x; q, t, s) which replace theA n-series interpolation Macdonald polynomialsP *μ (x; q, t) for theBC n-type root system. For these polynomials we prove an integral representation, a combinatorial formula, Pieri rules, Cauchy identity, and we also show that they do not satisfy any rationalq-difference equation. Ass → ∞ the polynomialsP *μ (x; q, t, s) becomeP *μ (x; q, t). We also prove a binomial formula for 6-parametric Koornwinder polynomials.

A Randomq, t-Hook Walk and a Sum of Pieri Coefficients

This work deals with the identityBμ(q, t)=∑ν→μcμν(q, t), whereBμ(q, t) denotes the biexponent generator of a partitionμ. That is,Bμ(q, t)=∑s∈μqa′(s)tl′(s), witha′(s) andl′(s) the co-arm and co-leg of the lattice squaresinμ. The coefficientscμν(q, t) are closely related to certain rational functions occuring in one of the Pieri rules for the Macdonald polynomials and the symbolν→μis used to indicate that the sum is over partitionsνwhich immediately precedeμin the Young lattice. This identity has an indirect manipulatorial proof involving a number of deep identities established by Macdonald. We show here that it may be given an elementary probabilistic proof by a mechanism which emulates the Greene–Nijehuis–Wilf proof of the hook formula.

On the Multiplication of Schubert Polynomials

Finding a combinatorial rule for the multiplication of Schubert polynomials is a long standing problem. In this paper we give a combinatorial proof of the extended Pieri rule as conjectured by N. Bergeron and S. Billey, which says how to multiply a Schubert polynomial by a complete or elementary symmetric polynomial, and describe some observations in the direction of a general rule.

Macdonald's evaluation conjectures and difference Fourier transform

This paper contains the proof of Macdonald's duality and evaluation conjectures, the definition of the difference Fourier transform, the recurrence theorem generalizing Pieri rules, and the action of GL(2,Z) on the Macdonald polynomials at roots of unity.

Factorizations of Pieri rules for Macdonald polynomials

We introduce a heuristic embedding of the Macdonald polynomials P μ ( x; q, t) into a family of polynomials indexed by lattice square diagrams. This embedding leads to recursions which may be viewed as a factorization of the Stanley-Macdonald (1988, 1989) Pieri rules and shed some light into their intricate nature. In this manner we can prove some conjectures concerning the coefficients K λμ ( q, t). In particular, when μ is a 2-row shape or a hook we show that the expression ∑ λƒ λK λμ(q, t) is a polynomial with nonnegative integer coefficients. The recursions obtained in these cases lead to a combinatorial interpretation of this polynomial as a q, t-enumerator of permutations. A new proof is also obtained of a Jacobi-Trudi identity for 2-row shapes recently obtained by Jing and Jósefiak. Some examples involving more general shapes are also included.

Counting paths in Young's lattice

Young's lattice is the lattice of partitions of integers, ordered by inclusion of diagrams. Standard Young tableaux can be represented as paths in Young's lattice that go up by one square at each step, and more general paths in Young's lattice correspond to more general kinds of tableaux. Using the theory of symmetric functions, in particular Pieri's rule for multiplying a Schur function by a complete symmetric function, we derive formulas for counting paths in Young's lattice that go up or down by horizontal or vertical strips. Our results are related to Richard Stanley's theory of differential posets in the special case of Young's lattice.

RC-Graphs and Schubert Polynomials

Using a formula of Billey, lockusch and Stanley, Fomin and Kirillov have introduced a new set of diagrams that encode the Schubert polynomials. We call theseobjects rc-graphs. We define and prove two variants of an algorithm for constructing the set of all rc-graphs for a given permutation. This construction makes many of the identities known for Schubert polynomials more apparent, and yields new ones. In particular, we give a new proof of Monk's rule using an insertion algorithm on rc-graphs. We conjecture two analogs of Pieri's rule for multiplying Schubert polynomials. We also extend the algorithm to generate the double Schubert polynomials.