A ‘Second Orthogonality Relation’ for Characters of Brauer Algebras

In a first course in representation theory , one usually learns that there are two important relations for characters of a finite group , the first orthogonality relation and the second orthogonality relation . When one moves on to study algebras which are not necessarily group algebras it is not clear , a priori , that the study of characters would be fruitful , and one encounters various problems in developing a theory analogous to that for finite groups . It is true , however , that the characters of algebras , such as the Iwahori – Hecke algebras , the Brauer algebras and the Birman – Wenzl algebras , are well-defined and important . An analogue of the first orthogonality relation for characters of such algebras is understood and appears in [4] . In the Appendix of this paper , we show that the second orthogonality relation for characters makes sense for split semisimple algebras (although it is no longer an orthogonality relation) . This paper is concerned with the ‘second orthogonality relation’ for the Brauer algebras . We derive this relation explicitly . After a talk on the results in Sections 1 and 2 of this paper at University of Bordeaux I , R . Stanley sketched an alternate proof of the results in Section 1 using the combinatorial interpretation of the characters of the Brauer algebra and tools from his paper [11] . Here we present the proof of Stanley’s as a theorem purely about the combinatorial rule for the characters of the Brauer algebras . Then we study the naturally occurring weight space representations of the Brauer algebra . Putting the three facets together , we are able to give a new derivation of the irreducible characters of the Brauer algebras . I would like to mention , here in the Introduction so that it gets noticed , that I have been unable to compute the second relation for characters explicitly in either the case of the Iwahori – Hecke algebra of type A or the case of the Birman – Wenzl algebra . Computing these relations could be useful in the study of representations of quantum groups and / or q -dif ferential posets and / or q -Hermite polynomials . This paper is organized as follows . In the Appendix we give an argument that there is an analogue of the second orthogonality relation for characters of a finite group in the case of any finitedimensional algebra with a non-degenerate trace form (in particular , a split semisimple algebra) . The purpose of this is to show that the study of the second relation for characters makes sense for the case of the Brauer algebra . We have put this material in an Appendix as it is primarily algebraic in nature and is not needed in the rest of paper . In Section 2 we derive by an enumerative argument the explicit form of the second relation for characters of the Brauer algebra . It is interesting to note that the formulas can be expressed in terms of products of certain Hermite polynomials . In Section 3 we give an application of our result of Section 2 to Weyl group symmetric functions of types B , C and D . Diaconis and Shahshahani [5] have also applied these results in their study of the eigenvalues of random orthogonal and symplectic matrices . In Section 4 we present Stanley’s proof of analogous ‘second relation’ formulas for certain numbers determined by up – down border strip tableaux . In Section 5 we show that the ‘second orthogonality relations’ of Sections 2 and 4 685