A note on Cartan matrices for symmetric groups

Using generating functions, a very simple explicit formula for the determinants of the p-Cartan matrices of symmetric groups is given. Our method works also when p is a composite number.Received: 5 September 2001     

On graded Cartan invariants of symmetric groups and Hecke algebras

We consider graded Cartan matrices of the symmetric groups and the Iwahori-Hecke algebras of type A at roots of unity. These matrices are \({\mathbb {Z}}[v,v^{-1}]\)-valued and may also be interpreted as Gram matrices of the Shapovalov form on sums of weight spaces of a basic representation of an affine quantum group. We present a conjecture predicting the invariant factors of these matrices and give evidence for the conjecture by proving its implications under a localization and certain specializations of the ring \({\mathbb {Z}}[v,v^{-1}]\). This proves and generalizes a conjecture of Ando-Suzuki-Yamada on the invariants of these matrices over \({\mathbb {Q}}[v,v^{-1}]\) and also generalizes the first author’s recent proof of the Külshammer-Olsson-Robinson conjecture over \({\mathbb {Z}}\).     

On defect groups for generalized blocks of the symmetric group

In a paper of 2003, Külshammer, Olsson and Robinson defined-blocks for the symmetric groups, where >1 is an arbitraryinteger. In this paper, we give a definition for the defectgroup of the principal -block. We then check that, in the Abeliancase, we have an analogue of one of Broué's conjectures.     

Generalized tangent representations for groups and symmetric spaces of matrices

Any two representations of dimensions n resp. r of a given group G allow the construction of a third representation φ in the space of rectangular n × r matrices K^n,r over the same ground field K. The φ-semidirect product of K^n,r and G then has (n + r) dimensional representation. The inhomogenizations of G and in case of matrix Lie groups G the tangent groups are special cases of this construction. The contragredient as well as the Lie algebraical versions of these results are included. In the final section the construction is generalized to symmetric spaces and their local algebraical structures, the Lie triples, by defining semidirect products resp. semidirect sums with respect to a representation     

On realization of fusion rings from generalized Cartan matrices

The Casimir element of a fusion ring (R, B) gives rise to the so called Casimir matrix C of (R, B). This enables us to construct a generalized Cartan matrix D-C in the sense of Kac for a suitable diagonal matrix D. In this paper, we study some elementary properties of the Casimir matrix C and use them to realize certain fusion rings from the generalized Cartan matrix D-C of finite (resp. affine) type. It turns out that there exists a fusion ring with D-C being of finite (resp. affine) type if and only if D-C has only the form A₂ (resp. A₁⁽¹⁾. We also realize all fusion rings with D-C being a particular generalized Cartan matrix of indefinite type.     

Tensor square of the basic spin representations of Schur covering groups for the symmetric groups

Journal of Algebraic Combinatorics - We consider the tensor square of the basic spin representations of Schur covering groups $$widetilde{S_n}$$ and $$widetilde{S_n^{'}}$$ for the symmetric...     

Cartan matrices for restricted Lie algebras

Consider a finite dimensional restricted Lie algebra over a field of prime characteristic. Each linear form on this Lie algebra defines a finite dimensional quotient of its universal enveloping algebra, called a reduced enveloping algebra. This leads to a Cartan matrix recording the multiplicities as composition factors of the simple modules in the projective indecomposable modules for such a reduced enveloping algebra. In this paper we show how to compare such Cartan matrices belonging to distinct linear forms. As an application we rederive and generalise the reciprocity formula first discovered by Humphreys for Lie algebras of reductive groups. For simple Lie algebras of Cartan type we see, for example, that the Cartan matrices for linear forms of non-positive height are submatrices of the Cartan matrix for the zero linear form.     

Linear bounds for the normal covering number of the symmetric and alternating groups

Monatshefte für Mathematik - The normal covering number $$gamma (G)$$ of a finite, non-cyclic group G is the minimum number of proper subgroups such that each element of G lies in some...     

On bounds for eigenvalues of real symmetric matrices

Let A=(a_ij) be a real symmetric matrix of order n. We characterize all nonnegative vectors x=(x₁,...,x_n) and y=(y₁,...,y_n) such that any real symmetric matrix B=(b_ij), with b_ij=a_ij, i≠jhas its eigenvalues in the union of the intervals [b_ij?y_i, b_ij+ x_i]. Moreover, given such a set of intervals, we derive better bounds for the eigenvalues of B using the 2n quantities {b_ii?y, b_ii+x_i}, i=1,..., n.     

On the stability of Cartan embeddings of compact symmetric spaces

Dedicated to Professor Tadashi Nagano on his sixtieth birthday     

On alternating and symmetric groups as Galois groups

Fix an integern≧3. We show that the alternating groupA _n appears as Galois group over any Hilbertian field of characteristic different from 2. In characteristic 2, we prove the same whenn is odd. We show that any quadratic extension of Hilbertian fields of characteristic different from 2 can be embedded in anS _n-extension (i.e. a Galois extension with the symmetric groupS _n as Galois group). Forn≠6, it will follow thatA _n has the so-called GAR-property over any field of characteristic different from 2. Finally, we show that any polynomialf=X ⁿ+…+a₁X+a₀ with coefficients in a Hilbertian fieldK whose characteristic doesn’t dividen(n-1) can be changed into anS _n-polynomialf ^* (i.e the Galois group off ^* overK Gal(f ^*, K), isS _n) by a suitable replacement of the last two coefficienta ₀ anda ₁. These results are all shown using the Newton polygon. The author acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2000-00114, GTEM.     

On quasiconvex hulls in symmetric matrices

In this paper we study the quasiconvex hull of compact sets of symmetric 2×2 matrices. We are interested in situations where the quasiconvex hull can be separated into smaller independent pieces. Our main result is a geometric criterion which is sufficient for the quasiconvex hull of the union of two compact sets K₁K₂ to separate in the sense that . The key point in the proof is a kind of directional maximum principle for second order elliptic equations in the plane in non-divergence form with measurable coefficients.     

On the decomposition map for symmetric groups

Let R ^d be the ℤ-module generated by the irreducible characters of the symmetric group . We determine bases for the kernel of the decomposition map. It is known that R ^d ⊗ _ℤ F is isomorphic to the radical quotient of the Solomon descent algebra when F is a field of characteristic zero. We show that when F has prime characteristic and I _br ^d is the kernel of the decomposition map for prime p then R ^d /I _br ^d ⊗ _ℤ F is isomorphic to the radical quotient of the p-modular Solomon descent algebra. To the memory of Manfred Schocker.