We derive a new expression for the supersymmetric Schur polynomials sλ(x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m|n) and gives rise to a determinantal formula for sλ(x/y). In the second part, we use this determinantal formula to derive new expressions for the dimension and superdimension of covariant representations Vλ of the Lie superalgebr...

We prove a determinantal identity concerning Schur functions for 2staircase diagrams λ = (ln+l, ln, l(n−1)+l′, l(n−1), . . . , l+l, l, l, 0). When l = 1 and l = 0 these functions are related to the partition function of the 6-vertex model at the combinatorial point and hence to enumerations of Alternating Sign Matrices. A consequence of our result is an identity concerning the doubly-refined nu...

We introduce the notion of the cutting strip of an outside decomposition of a skew shape, and show that cutting strips are in one-to-one correspondence with outside decompositions for a given skew shape. Outside decompositions are introduced by Hamel and Goulden and are used to give an identity for the skew Schur function that unifies the determinantal expressions for the skew Schur functions i...

We give lattice path proofs of determinantal formulas for orthosymplectic characters. We use the spo(2m, n)-tableaux introduced by Benkart, Shader and Ram, which have both a semistandard symplectic part and a rowstrict part. We obtain orthosymplectic Jacob-Trudi identities and an orthosymplectic Giambelli identity by associating spo(2m, n)-tableaux to certain families of nonintersecting lattice...

Recently Okada defined algebraically ninth variation skew Q-functions, in parallel to Macdonald’s Schur functions. Here we introduce a shifted tableaux definition of these and prove by means non-intersecting lattice path model Pfaffian outside decomposition result the form version Hamel’s identity. As corollaries this derive identities generalising those Józefiak–Pragacz, Nimmo, most recently O...

We prove an identity relating the permanent of a rank $2$ matrix and determinants its Hadamard powers. When viewed in right way, resulting formula looks strikingly similar to Carlitz Levine, suggesting possibility that these are actually special cases some more general (or class identities) connecting permanents determinants. The proof combines basic facts from theory symmetric functions with a...

We show that for any permutation w avoids a certain set of 13 patterns length 5 and 6, the Schubert polynomial \({\mathfrak {S}}_w\) can be expressed as determinant matrix elementary symmetric polynomials in manner similar to Jacobi–Trudi identity. For such w, this determinantal formula is equivalent (signed) subtraction-free expansion \(\mathfrak S_w\) basis standard monomials.

In this note we study standard and good determinantal schemes. We show that there exist arithmetically Cohen-Macaulay schemes that are not standard determinantal, and whose general hyperplane section is good determinantal. We prove that if a general hyperplane section of a scheme is standard (resp. good) determinantal, then the scheme is standard (resp. good) determinantal up to flat deformatio...