The algebra generated by the down and up operators on a differential or Ž . uniform partially ordered set poset encodes essential enumerative and structural properties of the poset. Motivated by the algebras generated by the down and up operators on posets, we introduce here a family of infinite-dimensional associative algebras called down᎐up algebras. We show that down᎐up algebras exhibit many Ž . of the important features of the universal enveloping algebra U ᒐ l of the Lie 2 algebra ᒐ ᒉ including a Poincare᎐BirkhoffᎏWitt type basis and a well-behaved 2 representation theory. We investigate the structure and representations of down᎐up algebras and focus especially on Verma modules, highest weight representations, and category O O modules for them. We calculate the exact expressions for all the weights, since that information has proven to be particularly useful in determining structural results about posets. ᮊ

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We define and characterize switching, an operation that takes two tableaux sharing a common border and``moves them through each other'' giving another such pair. Several authors, including James and Kerber, Remmel, Haiman, and Shimozono, have defined switching operations; however, each of their operations is somewhat different from the rest and each imposes a particular order on the switches that can occur. Our goal is to study switching in a general context, thereby showing that the previously defined operations are actually special instances of a single algorithm. The key observation is that switches can be performed in virtually any order without affecting the final outcome. Many known proofs concerning the jeu de taquin, Schur functions, tableaux, characters of representations, branching rules, and the Littlewood Richardson rule use essentially the same mechanism. Switching provides a common framework for interpreting these proofs. We relate Schu tzenberger's evacuation procedure to switching and in the process obtain further results concerning evacuation. We define reversal, an operation which extends evacuation to tableaux of arbitrary skew shape, and apply reversal and related mappings to give combinatorial proofs of various symmetries of Littlewood Richardson coefficients.

We present a uniform construction of level 1 perfect crystals B for all affine Lie algebras. We also introduce the notion of a crystal algebra and give an explicit description of its multiplication. This allows us to determine the energy function on B ⊗ B completely and thereby give a path realization of the basic representations at q = 0 in the homogeneous picture.

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Abstract. We investigate two-parameter quantum groups corresponding to the general linear and special linear Lie algebras gl n and sl n . We show that these quantum groups can be realized as Drinfel'd doubles of certain Hopf subalgebras with respect to Hopf pairings. Using the Hopf pairing, we construct a corresponding R-matrix and a quantum Casimir element. We discuss isomorphisms among these quantum groups and connections with multiparameter quantum groups.