The Schur polynomials satisfy many, many identities and there is a whole book about them. I think the easiest way is with the Vandermonde Determinant. $$s_{3,1,1}(a,b,c) = \frac{\left|\begin{array}{ccc} a^5 & b^5 & c^5 \\\\ a^2 & b^2 & c^2 \\\\ a & b & c\end{array} \right|}{(a - b)(b - c)(c - a)} = \left|\begin{array}{ccc} e_3 & e_2 & e_1 \\\\ e_2 & e_1 & e_0 \\\\ e_3 & e_2 & e_1\end{array} \right| = \left|\begin{array}{ccc} h_3 & h_2 & h_1 \\\\ h_2 & h_1 & h_0 \\\\ h_3 & h_2 & h_1\end{array} \right|$$ Where the $e_k$ are elementary symmetric polynomials and the $h_k$ are the homogeneous symmetric polynomials (the first is linear in each variable, the other isn't.) (3,1,1) is its own transpose.

Are there similar, easy to compute ways to find skew-Schur polynomials? For example, how to find $s_{(3,2,1)\backslash(1)}(x_1, x_2, x_3)$ - which is three squares along a diagonal the W penomino?

I know it is also possible to get (skew-)Schur polynomials as the matrix elements of a certain propagator acting on free fermions and using the Wick formula. For example, here.

The Schur polynomials satisfy many, many identities and there is a whole book about them. I think the easiest way is with the Vandermonde Determinant. $$s_{3,1,1}(a,b,c) = \frac{\left|\begin{array}{ccc} a^5 & b^5 & c^5 \\\\ a^2 & b^2 & c^2 \\\\ a & b & c\end{array} \right|}{(a - b)(b - c)(c - a)} = \left|\begin{array}{ccc} e_3 & e_2 & e_1 \\\\ e_2 & e_1 & e_0 \\\\ e_3 & e_2 & e_1\end{array} \right| = \left|\begin{array}{ccc} h_3 & h_2 & h_1 \\\\ h_2 & h_1 & h_0 \\\\ h_3 & h_2 & h_1\end{array} \right|$$ Where the $e_k$ are elementary symmetric polynomials and the $h_k$ are the homogeneous symmetric polynomials (the first is linear in each variable, the other isn't.) (3,1,1) is its own transpose.

Are there similar, easy to compute ways to find skew-Schur polynomials? For example, how to find $s_{(3,2,1)\backslash(1)}(x_1, x_2, x_3)$ - which is three squares along a diagonal the W penomino?

I know it is also possible to get (skew-)Schur polynomials as the matrix elements of a certain propagator acting on free fermions and using the Wick formula. For example, here.

The Schur polynomials satisfy many, many identities and there is a whole book about them. I think the easiest way is with the Vandermonde Determinant. $$s_{3,1,1}(a,b,c) = \frac{\left|\begin{array}{ccc} a^5 & b^5 & c^5 \\\\ a^2 & b^2 & c^2 \\\\ a & b & c\end{array} \right|}{(a - b)(b - c)(c - a)} = \left|\begin{array}{ccc} e_3 & e_2 & e_1 \\\\ e_2 & e_1 & e_0 \\\\ e_3 & e_2 & e_1\end{array} \right| = \left|\begin{array}{ccc} h_3 & h_2 & h_1 \\\\ h_2 & h_1 & h_0 \\\\ h_3 & h_2 & h_1\end{array} \right|$$ Where the $e_k$ are elementary symmetric polynomials and the $h_k$ are the homogeneous symmetric polynomials (the first is linear in each variable, the other isn't.) (3,1,1) is its own transpose.

Are there similar, easy to compute ways to find skew-Schur polynomials? For example, how to find $s_{(3,1,1)\backslash(2,1)}(x_1, x_2, x_3)$ - which is three squares along a diagonal?

I know it is also possible to get (skew-)Schur polynomials as the matrix elements of a certain propagator acting on free fermions and using the Wick formula. Any sources on that?

The Schur polynomials satisfy many, many identities and there is a whole book about them. I think the easiest way is with the Vandermonde Determinant. $$s_{3,1,1}(a,b,c) = \frac{\left|\begin{array}{ccc} a^5 & b^5 & c^5 \\\\ a^2 & b^2 & c^2 \\\\ a & b & c\end{array} \right|}{(a - b)(b - c)(c - a)} = \left|\begin{array}{ccc} e_3 & e_2 & e_1 \\\\ e_2 & e_1 & e_0 \\\\ e_3 & e_2 & e_1\end{array} \right| = \left|\begin{array}{ccc} h_3 & h_2 & h_1 \\\\ h_2 & h_1 & h_0 \\\\ h_3 & h_2 & h_1\end{array} \right|$$ Where the $e_k$ are elementary symmetric polynomials and the $h_k$ are the homogeneous symmetric polynomials (the first is linear in each variable, the other isn't.) (3,1,1) is its own transpose.

Are there similar, easy to compute ways to find skew-Schur polynomials? For example, how to find $s_{(3,2,1)\backslash(1)}(x_1, x_2, x_3)$ - which is three squares along a diagonal the W penomino?

I know it is also possible to get (skew-)Schur polynomials as the matrix elements of a certain propagator acting on free fermions and using the Wick formula. For example, here.