To each partition λ, we introduce a measure Sλ(x; t)sλ(y)/Zt where sλ is the Schur function and Sλ(x; t) is a generalization of the Schur function defined in [M] and Zt is a normalization constant. This measure, which we call the t-Schur measure, is a generalization of the Schur measure [O] and the shifted Schur measure studied by Tracy and Widom [TW3]. We prove that by a certain specialization...

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...

Schur Q-functions were originally introduced by Schur in relation to projective representations of the symmetric group and they can be defined combinatorially in terms of shifted tableaux. In this paper we describe planar decompositions of shifted tableaux into strips and use the shapes of these strips to generate pfaffi.ans and determinants that are equal to Schur Q-functions. As special cases...

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 fun...

We give a new explicit solution to the KP hierarchy. This is written in terms of Schur symmetric functions, and uses the known characterization of solutions to the KP hierarchy in terms of solutions to the Plücker relations. Our solution to the Pluc̈ker relations involves a countable set of variables for content, a combinatorial parameter for partitions (which themselves arise because they index...

We present a survey of recent developments of the Beilinson–Lusztig–MacPherson approach in the study of quantum gl n , infinitesimal quantum gl n , quantum gl ∞ and their associated q-Schur algebras, little q-Schur algebras and infinite q-Schur algebras. We also use the relationship between quantum gl ∞ and infinite q-Schur algebras to discuss their representations.

Using operators on compositions we develop further both the theory of quasisymmetric Schur functions and of noncommutative Schur functions. By establishing relations between these operators, we show that the posets of compositions arising from the right and left Pieri rules for noncommutative Schur functions can each be endowed with both the structure of dual graded graphs and dual filtered gra...

Many results involving Schur functions have analogues involving k-Schur functions. Standard strong marked tableaux play a role for k-Schur functions similar to the role standard Young tableaux play for Schur functions. We discuss results and conjectures toward an analogue of the hook-length formula.

We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satissed in many well-known associative algebras. As an application of our theory, we prove Schur-positivity and obtain generalized Littlewood-Richardson and Murnaghan-Nakayama rules for a large class of symmetric functions, including stable Schubert and Grothendieck polynomials.