A new generalization of Hermite’s reciprocity law

Abstract

Given a partition λ of n, the Schur functor S λ associates to any complex vector space V , a subspace S λ (V ) of V ⊗n . Hermite's reciprocity law, in terms of the Schur functor, states that S (p) S (q) (C 2 ) ≃ S (q) S (p) (C 2 ) . We extend this identity to many other identities of the type S λ S δ (C 2 ) ≃ Sµ Sǫ(C 2 ) .

Plethysms of symmetric functions and representations of SL 2 (C)

“…In particular, ∇ λ Sym E and ∇ µ Sym m E are isomorphic as representations of GL 2 (C) if and only if they are isomorphic as representations of SL 2 (C) and the degrees |λ| and m|µ| are equal. This is Theorem 3.1(ii) in [3], which our Proposition 3.6 extends.…”

“…By Corollary 2.15, cancelling the equal denominators [ 5 ) are equal up to a power of q. By Theorem 3.4(e) below, (8, 7, 2, 2) ∼ 5 5 (8,6,3). This example is generalized in Proposition 11.3.…”

“…Equivalently, t is bumpable in box (i, j) if and only if t (i,j) < , and increasing the entry of t in position (i, j) by 1 does not violate the semistandard condition. The following example shows the use of Definition 9.9 in the harder "only if" part of the proof of Theorem 9.5. are the unique tableaux in S 3 e (3 4 , 1 2 )/(2 2 ) for e ∈ {13, 14}, and the two tableaux in S 3 15 (3 4 , 1 2 )/(2 2 ) . Again the first tableau is t(λ/λ ).…”

“…Again the first tableau is t(λ/λ ). The second is its bump in position (4, 2), and the third and fourth both of weight |t(λ/λ )| + 2 are the bumps of the second in positions (3,2) and (4,2), respectively. Since the condition in (26) fails, s (3 4 ,1 2 )/(2 2 ) is not 3-irreducible.…”

Modular plethystic isomorphisms for two-dimensional linear groups

“…Obstructions to the conjugate partition isomorphism. Another classical result, due to King [Kin85, §4.2] (reproved as the main theorem in [CP16] and proved in a stronger version in [PW21, Theorem 1.5]), is that the representations ∇ (a+1,1 b ) Sym m+a E and ∇ (b+1,1 a ) Sym m+b E of SL 2 (C) are isomorphic for all m ∈ N 0 . By our final theorem, proved using the new modular invariant introduced in Definition 6.2, this isomorphism has, in general, no modular analogue, even after considering all possible dualities.…”