# Kernel of the map $mathbb{C}[G]^U to mathbb{C}[U^+]$

MathOverflow Asked by Jianrong Li on January 13, 2021

$$DeclareMathOperator{SL}{operatorname{SL}}$$Let $$G=SL_k$$ be the special linear group, $$U$$ the unipotent subgroup consisting of all lower unipotent triangular matrices, $$U^+$$ the unipotent subgroup consisting of all upper triangular matrices.

Denote by $$mathbb{C}[SL_k]^U$$ the $$U$$-invariant functions on $$SL_k$$. Consider the map $$varphi: mathbb{C}[SL_k]^U to mathbb{C}[U^+]$$ defined by restricting $$U$$-invariant functions on $$SL_k$$ to the subgroup $$U^+$$. As shown in the book by Fomin–Williams–Zelevinsky, Remark 6.5.7, this map is onto.

What is the kernel of $$varphi$$? Is it the ideal generated by leading principal minors?

Denote by $$widetilde{mathbb{C}[SL_k]^U}$$ the quotient of $$mathbb{C}[SL_k]^U$$ by the ideal generated by leading principal minors. Is it true that $$widetilde{mathbb{C}[SL_k]^U}$$ is isomorphic to $$mathbb{C}[U^+]$$ as algebras?

Thank you very much.

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