Symmetric Branching via Laurent Phenomenon Algebras

Abstract

We construct Laurent phenomenon structures on reductive branching algebras. For $A_{2n-1}\downarrow C_{n}$, $n\ge 2$, and the exceptional inclusions $D_4\downarrow G_2$, $E_6\downarrow F_4$, and $F_4\downarrow B_4$, we identify the branching algebras with upper Laurent phenomenon algebras, keeping the frozen coefficients polynomial. Degree-fibred categories of projective presentations give a common construction. It also gives cluster seeds of type $A_1$ for $B_3 \downarrow G_2$ and $G_2\downarrow A_2$. We give uniform sufficient conditions identifying a specialized Keel--Yu mirror algebra with an upper Laurent phenomenon algebra admitting a suitable binomial seed. The resulting theta basis is simultaneously adapted to the frozen boundary valuations. For $A_{2n-1}\downarrow C_{n}$ with $2\le n\le5$ and for the other exceptional inclusions, we obtain homogeneous theta bases parametrized by lattice points of explicit rational polyhedral cones, whose weight fibres compute every branching multiplicity. We also prove mutation invariance of finite upper bounds for LP patterns over a UFD.