Tableaux and insertion schemes for spinor representations of the orthogonal Lie algebraso(2r + 1, ℂ)

On Centralizer Algebras for Spin Representations

“…Observations to this extent have been made at the combinatorial level in several papers before, e.g. [1], [2]. However, the precise result we need is a little bit more subtle and does not immediately follow from the results above.…”

“…More specialized treatments can also be found in e.g. [1], [2] to name but a few. If V λ is an irreducible module with highest weight λ for Lie type B k , then…”

“…We will give an inductive procedure how to determine the structure of these algebras. To get the induction started, let us review the cases for P in (2) and Spin (3), which are well-known: P in (2): It is easy to check that P in (2) is isomorphic to the orthogonal group O(2): The two-fold covering of a circle is again a circle, which defines an isomorphism between Spin(2) and SO (2). This isomorphism extends to one between P in(2) and O(2).…”

“…[26], Lemma 7.5. Hence we have in general that C (2) 2 ∼ = C 3 and C (2) n+1 is isomorphic to a direct sum of Jones' basic construction for C (2) n−1 ⊂ C (2) n and a one-dimensional direct summand labeled by the Young diagram [n + 1] (see [5], [29]).…”

Insertion Scheme for the Classical Lie Algebras

“…In fact, there are tableaux other than Kashiwara-Nakashima which describe irreducible finite dimensional representations of the classical Lie algebras and for these tableaux, analogous works have been introduced by King (1975), De Concini (1979, ORDER REPRINTS Berele (1986), Sundaram (1990), Proctor (1990), and Benkart and Stroomer (1991). For the Lie superalgebra glðm; nÞ, crystal bases are realized as the set of skew tableaux by Benkart et al (2000) and an insertion scheme for this was introduced by Kang and Kwon (2001).…”