Some restriction coefficients for the trivial and sign representations

We use character polynomials to obtain a positive combinatorial interpretation of the multiplicity of the sign representation in irreducible polynomial representations of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>G</mml:mi> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℂ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> indexed by two-column and hook partitions. Our method also yields a positive combinatorial interpretation for the multiplicity of the trivial representation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> in an irreducible polynomial representation indexed by a hook partition.