Updown Robinson-Schensted correspondence through some algebraic varieties

Steinberg showed that the Robinson-Schensted (R-S) correspondence describes the relationship between two natural labelings of the irreducible components of an algebraic variety, sometimes called Steinberg's variety of triples. We give a similar interpretation, through some algebric varieties, of a variant of the R-S correspondence, conceived by Stanley and Sundaram and modified by Roby, which links the Brauer diagrams (also regarded as fixed-point-free involutions) with updown tableaux (also called oscillating tableaux). The talk will include a correction to a former preprint by the speaker, as well as some recent developments.