Jacobi–Trudi identity

The Jacobi–Trudi identity expresses a Schur polynomial asIn particular, $s_{(a,b)}=h_a{h}_b-h_{a+1}{h}_{b-1}$.

Here $[a]$ is the trivial character of $S_a$, $[0]$ is the induction-product unit, $[a]=0$ for $a<0$, and multiplication is induction of external tensor products. The Frobenius characteristic map turns the expression into the Jacobi–Trudi identity. A sign-reversing involution on intersecting lattice-path systems leaves precisely the nonintersecting systems corresponding to Semistandard Young tableaux, proving the determinant identity.