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I find new equations for Chow varieties, their secant varieties, and anadditional variety that arises in the study of depth 5 circuits by flatteningsand Koszul Young flattenings. This enables a new lower bound for symmetricborder rank of $x_1x_2\cdots x_d$ when $d$ is odd, and a lower bound on thesize of depth 5 circuits that compute the permanent.

Flattenings and Koszul Young flattenings arising in complexity theory