A Combinatorial Interpretation for the coefficients in the Kronecker Product s (n−p,p) ∗ s λ (Multiplicities in the Kronecker Product s (n−p,p) ∗ s λ )
Abstract
In this paper we give a combinatorial interpretation for the coefficient of
s
ν
in the Kronecker product
s
(n−p,p)
∗
s
λ
, where
λ=(
λ
1
,...,
λ
ℓ(λ)
)⊢n
, if
ℓ(λ)≥2p−1
or
λ
1
≥2p−1
; that is, if
λ
is not a partition inside the
2(p−1)×2(p−1)
square. For
λ
inside the square our combinatorial interpretation provides an upper bound for the coefficients. In general, we are able to combinatorially compute these coefficients for all
λ
when
n>(2p−2
)
2
. We use this combinatorial interpretation to give characterizations for multiplicity free Kronecker products. We have also obtained some formulas for special cases.