UC Davis

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].