Abstract: Given a monomial ideal $I$ in $R=K[x_1, ...x_n]$, the toric ring of $I$ is the $K$-subalgebra of $R$ generated by the monomial generators of $I$. A long-standing conjecture of White states that if $I$ is the ideal associated with a matroid, then the toric ring of $I$ is uniquely determined by the symmetric exchange relations among the generators of $I$. The conjecture was later extended to ideals associated with discrete polymatroids by Herzog and Hibi in 2002, and is only known to hold in certain cases, under additional assumptions.
In this talk, I will review the notion of matroids and discrete polymatroids and prove that the conjecture holds for symmetric polymatroidal ideals. This result is part of joint work with Alexandra Seceleanu, which was partially funded by an AWM Mentoring Travel Grant.
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