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On dimension formulas for g[(m|n) representations
Lie Theory and Its Applications in Physics V, 2004
We derive a new expression for the supersymmetric Schur polynomials s λ (x/y). The origin of this... more We derive a new expression for the supersymmetric Schur polynomials s λ (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 s λ (x/y). In the second part, we use this determinantal formula to derive new expressions for the dimension and superdimension of covariant representations V λ of the Lie superalgebra gl(m|n). In particular, we derive the t-dimension formula, giving a specialization of the character corresponding to the Z-grading of V λ. For a special choice of λ, the new t-dimension formula gives rise to a Hankel determinant identity.
Journal of Algebraic Combinatorics - J ALGEBR COMB, 2003
We derive a new formula for the supersymmetric Schur polynomial sλ(x/y). The origin of this formu... more We derive a new formula for the supersymmetric Schur polynomial sλ(x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m/n). In particular, we show how a character formula due to Kac and Wakimoto can be applied to covariant representations, leading to a new expression for sλ(x/y). This new expression gives rise to a determinantal formula for sλ(x/y). In particular, the denominator identity for gl(m/n) corresponds to a determinantal identity combining Cauchy's double alternant with Vandermonde's determinant. We provide a second and independent proof of the new determinantal formula by showing that it satisfies the four characteristic properties of supersymmetric Schur polynomials. A third and more direct proof ties up our formula with that of Sergeev-Pragacz.
Journal of Physics A: Mathematical and General, 2004
Let g be the Lie superalgebra gl(m|n). We show how to associate a gl(m|n) weight Λ to a composite... more Let g be the Lie superalgebra gl(m|n). We show how to associate a gl(m|n) weight Λ to a composite partition ν; µ with composite Young diagram F (ν; µ). Based upon the definition of critical representations, the notion of "critical composite partition" is introduced. It is shown that for critical composite partitions (subject to a technical restriction) the corresponding gl(m|n) representation V Λ is tame, so its character formula can be computed. This character is shown to coincide with the composite S-function s ν;µ (x/y).
Supersymmetric Schur functions and Lie superalgebra representations
It is shown how to associate to a highest weight Λ of the Lie superalgebra gl(m|n) a composite pa... more It is shown how to associate to a highest weight Λ of the Lie superalgebra gl(m|n) a composite partition ν; µ with composite Young diagram F (ν; µ). The corresponding supersymmetric Schur function sν;µ(x/y) is defined. However, it is known that this S-function does not always coincide with the character of the irreducible representation VΛ with highest weight Λ. Only for covariant, contravariant and typical representations the character and the S-function are known to coincide. Here, the notions of critical composite partitions and critical highest weights are considered. It is shown that for critical composite partitions (subject to a technical restriction) the corresponding gl(m|n) representation VΛ is tame, so its character can be computed. Also for this class of representations the character coincides with the composite supersymmetric S-function sν;µ(x/y). This extends considerably the classes of gl(m|n) representations for which the character can be computed by means of Sfunctions.
[
On dimension formulas for g[(m|n) representations
Lie Theory and Its Applications in Physics V, 2004
We derive a new expression for the supersymmetric Schur polynomials s λ (x/y). The origin of this... more We derive a new expression for the supersymmetric Schur polynomials s λ (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 s λ (x/y). In the second part, we use this determinantal formula to derive new expressions for the dimension and superdimension of covariant representations V λ of the Lie superalgebra gl(m|n). In particular, we derive the t-dimension formula, giving a specialization of the character corresponding to the Z-grading of V λ. For a special choice of λ, the new t-dimension formula gives rise to a Hankel determinant identity.
Journal of Algebraic Combinatorics - J ALGEBR COMB, 2003
We derive a new formula for the supersymmetric Schur polynomial sλ(x/y). The origin of this formu... more We derive a new formula for the supersymmetric Schur polynomial sλ(x/y). The origin of this formula goes back to representation theory of the Lie superalgebra gl(m/n). In particular, we show how a character formula due to Kac and Wakimoto can be applied to covariant representations, leading to a new expression for sλ(x/y). This new expression gives rise to a determinantal formula for sλ(x/y). In particular, the denominator identity for gl(m/n) corresponds to a determinantal identity combining Cauchy's double alternant with Vandermonde's determinant. We provide a second and independent proof of the new determinantal formula by showing that it satisfies the four characteristic properties of supersymmetric Schur polynomials. A third and more direct proof ties up our formula with that of Sergeev-Pragacz.
Journal of Physics A: Mathematical and General, 2004
Let g be the Lie superalgebra gl(m|n). We show how to associate a gl(m|n) weight Λ to a composite... more Let g be the Lie superalgebra gl(m|n). We show how to associate a gl(m|n) weight Λ to a composite partition ν; µ with composite Young diagram F (ν; µ). Based upon the definition of critical representations, the notion of "critical composite partition" is introduced. It is shown that for critical composite partitions (subject to a technical restriction) the corresponding gl(m|n) representation V Λ is tame, so its character formula can be computed. This character is shown to coincide with the composite S-function s ν;µ (x/y).
Supersymmetric Schur functions and Lie superalgebra representations
It is shown how to associate to a highest weight Λ of the Lie superalgebra gl(m|n) a composite pa... more It is shown how to associate to a highest weight Λ of the Lie superalgebra gl(m|n) a composite partition ν; µ with composite Young diagram F (ν; µ). The corresponding supersymmetric Schur function sν;µ(x/y) is defined. However, it is known that this S-function does not always coincide with the character of the irreducible representation VΛ with highest weight Λ. Only for covariant, contravariant and typical representations the character and the S-function are known to coincide. Here, the notions of critical composite partitions and critical highest weights are considered. It is shown that for critical composite partitions (subject to a technical restriction) the corresponding gl(m|n) representation VΛ is tame, so its character can be computed. Also for this class of representations the character coincides with the composite supersymmetric S-function sν;µ(x/y). This extends considerably the classes of gl(m|n) representations for which the character can be computed by means of Sfunctions.