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SIMPLE LIE SUPERALGEBRAS AND NONINTEGRABLE DISTRIBUTIONS IN CHARACTERISTIC p
, 2006
"... Abstract. Recently, Grozman and Leites returned to the original Cartan’s description of Lie algebras to interpret the Melikyan algebras (for p≤5) and several other littleknown simple Lie algebras over algebraically closed fields for p = 3 as subalgebras of Lie algebras of vector fields preserving n ..."
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Abstract. Recently, Grozman and Leites returned to the original Cartan’s description of Lie algebras to interpret the Melikyan algebras (for p≤5) and several other littleknown simple Lie algebras over algebraically closed fields for p = 3 as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by G(2), O(7), Sp(4) and Sp(10). The description was performed in terms of CartanTanakaShchepochkina prolongs using Shchepochkina’s algorithm and with the help of SuperLie package. Grozman and Leites also found two new series of simple Lie algebras. Here we apply the same method to distributions preserved by one of the two exceptional simple finite dimensional Lie superalgebras over C; for p = 3, we obtain a series of new simple Lie superalgebras and an exceptional one. In memory of Felix Aleksandrovich Berezin F. A. Berezin and supersymmetries are usually associated with physics. However, Lie superalgebras — infinitesimal supersymmetries — appeared in topology at approximately the same time as the word “spin ” appeared in physics and it were these examples that Berezin first had in mind.
A unified formula for Steenrod operations in flag manifolds
, 2005
"... A unified formula for Steenrod operations in flag manifolds ..."
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A unified formula for Steenrod operations in flag manifolds
The cohomology of the Steenrod algebra and representations of the general linear groups
 Trans. Amer. Math. Soc
"... ABSTRACT. Let Trk be the algebraic transfer that maps from the coinvariants of certain GLkrepresentation to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer trk: n~((BVk)+)7 11'~(8°). It has been shown that the ..."
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ABSTRACT. Let Trk be the algebraic transfer that maps from the coinvariants of certain GLkrepresentation to the cohomology of the Steenrod algebra. This transfer was defined by W. Singer as an algebraic version of the geometrical transfer trk: n~((BVk)+)7 11'~(8°). It has been shown that the algebraic transfer is highly nontrivial, more precisely, that Trk is an isomorphism for k = 1, 2, 3 and that Tr = ffikTrk is a homomorphism of algebras. In this paper, we first recognize the phenomenon that if we start from any degree d, and apply Sq0 repeatedly at most (k 2) times, then we get into the region, in which all the iterated squaring operations are isomorphisms on the coinvariants of the GLkrepresentation. As a consequence, every finite Sq0family in the coinvariants has at most (k 2) non zero elements. Two applications are exploited. The first main theorem is that Trk is not an isomorphism for k 2: 5. Furthermore, Trk is not an isomorphism in infinitely many degrees for each k> 5. We also show that if Tre detects a nonzero element in certain de
Computing Cocycles on Simplicial Complexes
"... In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares [20,21], in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we ..."
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In this note, working in the context of simplicial sets [17], we give a detailed study of the complexity for computing chain level Steenrod squares [20,21], in terms of the number of face operators required. This analysis is based on the combinatorial formulation given in [5]. As an application, we give here an algorithm for computing cupi products over integers on a simplicial complex at chain level. 1
On the homology of elementary Abelian groups as modules over the Steenrod algebra
 J. Pure Appl. Algebra
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Deformed diagonal harmonic polynomials for complex reflection groups
 In 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC
, 2011
"... Abstract. We introduce deformations of the space of (multidiagonal) harmonic polynomials for any finite complex reflection group of the form W = G(m, p, n), and give supporting evidence that this space seems to always be isomorphic, as a graded Wmodule, to the undeformed version. Résumé. Nous intr ..."
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Abstract. We introduce deformations of the space of (multidiagonal) harmonic polynomials for any finite complex reflection group of the form W = G(m, p, n), and give supporting evidence that this space seems to always be isomorphic, as a graded Wmodule, to the undeformed version. Résumé. Nous introduisons une déformation de l’espace des polynômes harmoniques (multidiagonaux) pour tout groupe de réflexions complexes de la forme W = G(m, p, n), et soutenons l’hypothèse que cet espace est toujours isomorphe, en tant que Wmodule gradué, à l’espace d’origine.
Thiéry. Deformation of symmetric functions and the rational Steenrod algebra
 In Invariant theory in all characteristics, volume 35 of CRM Proc. Lecture
"... Abstract. In 1999, Reg Wood conjectured that the quotient of Q[x1,..., xn] by the action of the rational Steenrod algebra is a graded regular representation of the symmetric group Sn. As pointed out by Reg Wood, the analog of this statement is a well known result when the rational Steenrod algebra i ..."
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Abstract. In 1999, Reg Wood conjectured that the quotient of Q[x1,..., xn] by the action of the rational Steenrod algebra is a graded regular representation of the symmetric group Sn. As pointed out by Reg Wood, the analog of this statement is a well known result when the rational Steenrod algebra is replaced by the ring of symmetric functions; actually, much more is known about the structure of the quotient in this case. We introduce a noncommutative qdeformation of the ring of symmetric functions, which specializes at q = 1 to the rational Steenrod algebra. We use this formalism to obtain some partial results. Finally, we describe several conjectures based on an extensive computer exploration. In particular, we extend Reg Wood’s conjecture to q formal and to any q ∈ C not of the form
HARMONICS FOR DEFORMED STEENROD OPERATORS
, 812
"... Abstract. We explore in this paper the spaces of common zeros of several deformations of Steenrod operators. Contents ..."
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Abstract. We explore in this paper the spaces of common zeros of several deformations of Steenrod operators. Contents
SOME FORMULAS FOR THE ACTION OF STEENROD POWERS ON
"... Abstract. In this study we give some formulas for the action of Steenrod powers on certain monomials and some polynomials having these monomials as a factor in the polynomial algebra P (n) = Zp [x1; : : : ; xn], deg (xi) = 2, i = 1; : : : ; n and p is an odd prime. We also give some new family of ..."
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Abstract. In this study we give some formulas for the action of Steenrod powers on certain monomials and some polynomials having these monomials as a factor in the polynomial algebra P (n) = Zp [x1; : : : ; xn], deg (xi) = 2, i = 1; : : : ; n and p is an odd prime. We also give some new family of hit polynomials.