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Register a new userRepresentation theory of symmetric groups and the strong Lefschetz property
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We investigate the structure and properties of an Artinian monomial complete intersection quotient $A(n,d)=mathbf{k} [x_{1}, ldots, x_{n}] big / (x_{1}^{d}, ldots, x_{n}^d)$. We construct explicit homogeneous bases of $A(n,d)$ that are compatible with the $S_{n}$-module structure for $n=3$, all exponents $d ge 3$ and all homogeneous degrees $j ge 0$. Moreover, we derive the multiplicity formulas, both in recursive form and in closed form, for each irreducible component appearing in the $S_{3}$-module decomposition of homogeneous subspaces. 4, 5$.
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For the coinvariant rings of finite Coxeter groups of types other than H$_4$, we show that a homogeneous element of degree one is a strong Lefschetz element if and only if it is not fixed by any reflections. We also give the necessary and sufficient condition for strong Lefschetz elements in the invariant subrings of the coinvariant rings of Weyl groups.
In this paper we consider symmetric powers representation and exterior powers representation of finite groups, which generated by the representation which has finite dimension over the complex field. We calculate the multiplicity of irreducible component of two representations of some representation by using a character theory of representation and a pre-lambda-ring, for example, the regular representation.
Geometric symmetry induces symmetries of function spaces, and the latter yields a clue to global analysis via representation theory. In this note we summarize recent developments on the general theory about how geometric conditions affect representation theoretic properties on function spaces, with focus on multiplicities and spectrum.
In this paper we study invariant rings arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form $K[U]^{Gamma}$ where $Gamma$ is a product of general linear groups over a field $K$ of characteristic zero, and $U$ is a finite dimensional rational representation of $Gamma$. We will calculate the Hilbert series of such rings using the representation theory of the symmetric groups and Schur-Weyl duality. We focus on the case where $U=text{End}(W^{oplus k})$ and $Gamma = text{GL}(W)$ and on the case where $U=text{End}(Votimes W)$ and $Gamma = text{GL}(V)times text{GL}(W)$, though the methods introduced here can also be applied in more general framework. For the two aforementioned cases we calculate the Hilbert function of the ring of invariants in terms of Littlewood-Richardson and Kronecker coefficients. When the vector spaces are of dimension 2 we also give an explicit calculation of this Hilbert series.
We classify the irreducible projective representations of symmetric and alternating groups of minimal possible and second minimal possible dimensions, and get a lower bound for the third minimal dimension. On the way we obtain some new results on branching which might be of independent interest.
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