A Quick (ab initio and non-adhoc) Construction of Irreducible
Representations of Symmetric & General Linear Groups
Abstract: The description of the IrReps (short for `Irreducible Representations') of both classes of groups (Symmetric Groups \S_k of order k!, as also the GL_n) involves the ``Lattice of all Partitions''. I'll begin the talk with some associated `magical numerologies' that are not too well-known (concerning the dimensions of those IrReps, where the multitude of formulae can only be explained in terms of various somewhat `deep' results: one of those being the `Tensor Decompostion Theorem'). )From there we go on to show how the entire old edifice of these IrReps can be introduced in a fairly naive manner. The ideas go back to exactly a Century ago (I.Schur's Thesis); \&are most basic to many a diverse Mathematics its Applications, but still need wider dissemination; while our constructions are entirely elementary, some of the key steps in the `proof' need a small dosage of Lie Theory (but only to the extent of what's already widely known -- ia a relative sense). [Yet for most `applications purposes' the said proof \& the said expertise in Lie Theory is quite un-necessary.]
Thus our emphasis is on quick recovery, and a global view, of some very fascinating mathematics (which happens to be too crucial for many users). In the 1-hour talk we shall need to restrict attention to the former class \S_k -- where things are less widely understood; but shall also endeavor to emphasize a basic analogy between the two parallel constrctions. At least getting at bare hints of the rationale for the deep (`magical') numerologies introduced at the outset, ought to be possible; in proper detail, what is involved is the very `elaborate' theory of ``Young Tableaus'' -- which is the backbone of many newer directions in Combinatorics; literally hundreds of papers are appearing on them each year, and yet it safe to say that the subject is surrounded by a vast amount of `mysticism'. Our `elementary constructions' have the advantage of making much of the existing ``voodoo'' of Tableaux Theory appear (again) as a `Rational Science' (= non-adhoc Mathematics).