Abstract:
We introduce the immersion poset
$$({\mathcal {P}}(n), \leqslant _I)$$
(
P
(
n
)
,
⩽
I
)
on partitions, defined by
$$\lambda \leqslant _I \mu $$
λ
⩽
I
μ
if and only if
$$s_\mu (x_1, \ldots , x_N) - s_\lambda (x_1, \ldots , x_N)$$
s
μ
(
x
1
,
…
,
x
N
)
-
s
λ
(
x
1
,
…
,
x
N
)
is monomial-positive. Relations in the immersion poset determine when irreducible polynomial representations of
$$GL_N({\mathbb {C}})$$
G
L
N
(
C
)
form an immersion pair, as defined by Prasad and Raghunathan [7]. We develop injections
$$\textsf{SSYT}(\lambda ,
u ) \hookrightarrow \textsf{SSYT}(\mu ,
u )$$
SSYT
(
λ
,
ν
)
↪
SSYT
(
μ
,
ν
)
on semistandard Young tableaux given constraints on the shape of
$$\lambda $$
λ
, and present results on immersion relations among hook and two column partitions. The standard immersion poset
$$({\mathcal {P}}(n), \leqslant _{std})$$
(
P
(
n
)
,
⩽
std
)
is a refinement of the immersion poset, defined by
$$\lambda \leqslant _{std} \mu $$
λ
⩽
std
μ
if and only if
$$\lambda \leqslant _D \mu $$
λ
⩽
D
μ
in dominance order and
$$f^\lambda \leqslant f^\mu $$
f
λ
⩽
f
μ
, where
$$f^
u $$
f
ν
is the number of standard Young tableaux of shape
$$
u $$
ν
. We classify maximal elements of certain shapes in the standard immersion poset using the hook length formula. Finally, we prove Schur-positivity of power sum symmetric functions on conjectured lower intervals in the immersion poset, addressing questions posed by Sundaram [12].
