http://idea.library.drexel.edu/handle/1860/1173 Search the Channel search http://idea.library.drexel.edu/simple-search http://idea.library.drexel.edu/handle/1860/2731 Title: Conservative dilations of dissipative multidimensional systems: the commutative and non-commutative settings <br/> <br/>Authors: Ball, Joseph A.; Kaliuzhnyi-Verbovetskyi, Dmitry S. <br/> <br/>Abstract: We establish the existence of conservative dilations for various types of dissipative non-commutative N-dimensional (N-D) systems. As a corollary, a criterion of existence of conservative dilations for corresponding dissipative commutative N-D systems is obtained. We point out the cases where this criterion is always fulfilled, and the cases where it is not always fulfilled. http://idea.library.drexel.edu/handle/1860/2700 Title: Two variable orthogonal polynomials on the bicircle and structured matrices <br/> <br/>Authors: Geronimo, Jeffrey S.; Woerdeman, Hugo <br/> <br/>Abstract: We consider bivariate polynomials orthogonal on the bicircle with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between the polynomials of different degrees. These formulas link the orthogonal polynomials constructed using the lexicographical ordering with those constructed using the reverse lexicographical ordering. Relations between the coefficients in the recurrence formulas are derived and used to give necessary and sufficient conditions for the existence of a positive linear functional. These results are then used to construct a class of two variable measures supported on the bicircle that are given by one over the magnitude squared of a stable polynomial. Applications to Fej´er–Riesz factorization are also given. http://idea.library.drexel.edu/handle/1860/1947 Title: Schur function analogs for a filtration of the symmetric function space <br/> <br/>Authors: Morse, Jennifer; Lapointe, Luc <br/> <br/>Abstract: We consider a filtration of the symmetric function space given by At(k) , the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than k. We introduce symmetric functions called the k-Schur functions, providing an analog for the Schur functions in the subspaces At(k) . We prove several properties for the k-Schur functions including that they form a basis for these subspaces that reduces to the Schur basis when k is large. We also show that the connection coefficients for the k-Schur function basis with the Macdonald polynomials belonging to At(k) are polynomials in q and t with integral coefficients. In fact, we conjecture that these integral coefficients are actually positive, and give several other conjectures generalizing Schur function theory. http://idea.library.drexel.edu/handle/1860/1946 Title: Order ideals in weak subposets of Young’s lattice and associated unimodality conjectures <br/> <br/>Authors: Morse, Jennifer; Lapointe, Luc <br/> <br/>Abstract: The k-Young lattice Y k is a weak subposet of the Young lattice containing partitions whose first part is bounded by an integer k > 0. The Y k poset was introduced in connection with generalized Schur functions and later shown to be isomorphic to the weak order on the quotient of the affine symmetric group ˜ Sk+1 by a maximal parabolic subgroup. We prove a number of properties for Y k including that the covering relation is preserved when elements are translated by rectangular partitions with hook-length k. We highlight the order ideal generated by an m× n rectangular shape. This order ideal, Lk(m, n), reduces to L(m, n) for large k, and we prove it is isomorphic to the induced subposet of L(m, n) whose vertex set is restricted to elements with no more than k−m+1 parts smaller than m. We provide explicit formulas for the number of elements and the rank-generating function of Lk(m, n). We conclude with unimodality conjectures involving q-binomial coefficients and discuss how implications connect to recent work on sieved q-binomial coefficients. http://idea.library.drexel.edu/handle/1860/1945 Title: Affine insertion and Pieri rules for the affine Grassmannian <br/> <br/>Authors: Morse, Jennifer; Lam, Thomas; Lapointe, Luc; Shimozono, Mark <br/> <br/>Abstract: We study combinatorial aspects of the Schubert calculus of the affine Grassmannian Gr associated with SL(n, C). Our main results are: • Pieri rules for the Schubert bases of H∗(Gr) and H∗(Gr), which expresses the product of a special Schubert class and an arbitrary Schubert class in terms of Schubert classes. • A new combinatorial definition for k-Schur functions, which represent the Schubert basis of H∗(Gr). • A combinatorial interpretation of the pairing H∗(Gr)×H∗(Gr) --> Z. These results are obtained by interpreting the Schubert bases of Gr combinatorially as generating functions of objects we call strong and weak tableaux, which are respectively defined using the strong and weak orders on the affine symmetric group. We define a bijection called affine insertion, generalizing the Robinson-Schensted Knuth correspondence, which sends certain biwords to pairs of tableaux of the same shape, one strong and one weak. Affine insertion offers a duality between the weak and strong orders which does not seem to have been noticed previously. Our cohomology Pieri rule conjecturally extends to the affine flag manifold, and we give a series of related combinatorial conjectures. http://idea.library.drexel.edu/handle/1860/1944 Title: Determinantal expressions for Macdonald polynomials <br/> <br/>Authors: Morse, Jennifer; Lapointe, Luc; Lascoux, Alain <br/> <br/>Abstract: We show that the action of classical operators associated to the Macdonald polynomials on the basis of Schur functions, Sλ [X(t − 1)/(q − 1)], can be reduced to addition in λ−rings. This provides explicit formulas for the Macdonald polynomials expanded in this basis as well as in the ordinary Schur basis, Sλ[X], and the monomial basis, mλ[X]. http://idea.library.drexel.edu/handle/1860/1943 Title: The distributions of the entries of Young tableaux <br/> <br/>Authors: Morse, Jennifer <br/> <br/>Abstract: Let T be a standard Young tableau of shape λ ⊢ k. We show that the probability that a randomly chosen Young tableau of n cells contains T as a subtableau is, in the limit n → ∞, equal to f_/k!, where f_ is the number of all tableaux of shape λ. In other words, the probability that a large tableau contains T is equal to the number of tableaux whose shape is that of T , divided by k!. We give several applications, to the probabilities that a set of prescribed entries will appear in a set of prescribed cells of a tableau, and to the probabilities that subtableaux of given shapes will occur. Our argument rests on a notion of quasirandomness of families of permutations, and we give sufficient conditions for this to hold. http://idea.library.drexel.edu/handle/1860/1942 Title: Tableau atoms and a new Macdonald positivity conjecture <br/> <br/>Authors: Morse, Jennifer; Lapointe, Luc; Lascoux, Alain <br/> <br/>Abstract: Let A be the space of symmetric functions and Vk be the subspace spanned by the modified Schur functions {Sy[X/(1 − t)]}1k. We introduce a new family of symmetric polynomials, {A(k) [X; t]}1<k, constructed from sums of tableaux using the charge statistic. We conjecture that the polynomials Ay(k) [X; t] form a basis for Vk and that the Macdonald polynomials indexed by partitions whose first part is not larger than k expand positively in terms of our polynomials. A proof of this conjecture would not only imply the Macdonald positivity conjecture, but would substantially refine it. Our construction of the Ay(k) [X; t] relies on the use of tableaux combinatorics and yields various properties and conjectures on the nature of these polynomials. Another important development following from our investigation is that the Ay(k) [X; t] seem to play the same role for Vk as the Schur functions do for . In particular, this has led us to the discovery of many generalizations of properties held by the Schur functions, such as Pieri and Littlewood-Richardson type coefficients. http://idea.library.drexel.edu/handle/1860/1941 Title: Schur function identities, their t-analogs, and k-Schur irreducibility <br/> <br/>Authors: Morse, Jennifer; Lapointe, Luc <br/> <br/>Abstract: We obtain general identities for the product of two Schur functions in the case where one of the functions is indexed by a rectangular partition, and give their t-analogs using vertex operators. We study subspaces forming a filtration for the symmetric function space that lends itself to generalizing the theory of Schur functions and also provides a convenient environment for studying the Macdonald polynomials. We use our identities to prove that the vertex operators leave such subspaces invariant. We finish by showing that these operators act simply on the k-Schur functions, thus leading to a concept of irreducibility for these functions. http://idea.library.drexel.edu/handle/1860/1940 Title: Quantum cohomology and the k-Schur basis <br/> <br/>Authors: Morse, Jennifer; Lapointe, Luc <br/> <br/>Abstract: We prove that structure constants related to Hecke algebras at roots of unity are special cases of k-Littlewood-Richardson coefficients associated to a product of k-Schur functions. As a consequence, both the 3- point Gromov-Witten invariants appearing in the quantum cohomology of the Grassmannian, and the fusion coefficients for the WZW conformal field theories associated to csu(ℓ) are shown to be k-Littlewood Richardson coefficients. From this, Mark Shimozono conjectured that the k-Schur functions form the Schubert basis for the homology of the loop Grassmannian, whereas k-Schur coproducts correspond to the integral cohomology of the loop Grassmannian. We introduce dual k-Schur functions defined on weights of k-tableaux that, given Shimozono’s conjecture, form the Schubert basis for the cohomology of the loop Grassmannian. We derive several properties of these functions that extend those of skew Schur functions.