scholarly journals LIE POWERS OF THE NATURAL MODULE FOR GL(2,K)

2011 ◽  
Vol 64 (1) ◽  
pp. 113-131
Author(s):  
K. Erdmann ◽  
M. Johnson
Keyword(s):  
Natural Module

scholarly journals A REMARK ON THE PERMUTATION REPRESENTATIONS AFFORDED BY THE EMBEDDINGS OF IN

2013 ◽  
Vol 89 (2) ◽  
pp. 331-336 ◽  
Author(s):  
SIMON GUEST ◽  
ANDREA PREVITALI ◽  
PABLO SPIGA
Keyword(s):  
Permutation Module ◽  
Natural Module ◽  
The Right

AbstractWe show that the permutation module over $ \mathbb{C} $ afforded by the action of ${\mathrm{Sp} }_{2m} ({2}^{f} )$ on its natural module is isomorphic to the permutation module over $ \mathbb{C} $ afforded by the action of ${\mathrm{Sp} }_{2m} ({2}^{f} )$ on the union of the right cosets of ${ \mathrm{O} }_{2m}^{+ } ({2}^{f} )$ and ${ \mathrm{O} }_{2m}^{- } ({2}^{f} )$.

Truncated symmetric powers and modular representations of GLn

1996 ◽  
Vol 119 (2) ◽  
pp. 231-242 ◽  
Author(s):  
Stephen Doty ◽  
Grant Walker
Keyword(s):  
General Linear Group ◽  
Schur Functions ◽  
Tensor Products ◽  
Modular Representation ◽  
Modular Representations ◽  
Modular Representation Theory ◽  
Simple Modules ◽  
Symmetric Powers ◽  
Natural Module ◽  
Natural Way

AbstractSeveral results are obtained relating to the modular representation theory of the general linear group GLn in the defining characteristic p > 0. In Section 1, embeddings of certain simple modules in symmetric powers of the natural module, or in tensor products of truncated symmetric powers, are constructed. In Section 2, cases are found where simple quotientsof Schur modules H0(λ) can be constructed by extending theidea of truncation to these modules in a natural way. In Section 3, the characters of those simple modules which can be constructed as twisted tensor products of truncated symmetric powers are expressed in terms of Schur functions.

scholarly journals The E-Cohomological Conley Index, Cup-Lengths and the Arnold Conjecture on T 2n

2019 ◽  
Vol 19 (3) ◽  
pp. 519-528 ◽  
Author(s):  
Maciej Starostka ◽  
Nils Waterstraat
Keyword(s):  
Lower Bound ◽  
Hilbert Spaces ◽  
Critical Points ◽  
Short Proof ◽  
Conley Index ◽  
Module Structure ◽  
Arnold Conjecture ◽  
Natural Module ◽  
The Way

Abstract We show that the E-cohomological Conley index, that was introduced by the first author recently, has a natural module structure. This yields a new cup-length and a lower bound for the number of critical points of functionals on Hilbert spaces. When applied to the setting of the Arnold conjecture, this paves the way to a short proof on tori, where it was first shown by C. Conley and E. Zehnder in 1983.

Shamanism and Cognitive Evolution (with comments)

2002 ◽  
Vol 12 (1) ◽  
pp. 71-101 ◽  
Author(s):  
Michael Winkelman
Keyword(s):  
Social Relations ◽  
Social Evolution ◽  
Mental States ◽  
Natural World ◽  
Altered States Of Consciousness ◽  
Altered States ◽  
Upper Palaeolithic ◽  
States Of Consciousness ◽  
Natural Module ◽  
Cave Art

Shamanic referents in Upper Palaeolithic cave art indicate its pivotal role in the Middle–Upper Palaeolithic transition. Etic models of shamanism derived from cross-cultural research help articulate the shamanic paradigm in cave art and explicate the role of shamanism in this transition. Shamanism is found cross-culturally in hunter-gatherer societies, constituting an ecological and psychosociobiological adaptation that reflects the ritual and cosmology of early modern humans. Shamanism played a role in cognitive and social evolution through production of analogical thought processes, visual symbolism and group-bonding rituals. Universals of shamanism are derived from innate modules, particularly the hominid ‘mimetic controller’ and music and dance. These induced altered states of consciousness, which produce physiological, cognitive, personal and social integration through integrative brain-processing. Shamanic altered states of consciousness have the cross-modal integration characteristic of the emergent features of Palaeolithic thought and facilitated adaptations to the ecological and social changes of the Upper Palaeolithic. Cross-modular integration of innate modules for inferring mental states (mind), and social relations (self/others), and understanding the natural world (classificatory schemas) produced the fundamental forms of trope (metaphor) that underlay analogical representation. These integrations also explain animism (mental and social modules applied to natural domains); totemism (natural module applied to social domain); and guardian spirit relations (natural module applied to self and mental domains).

scholarly journals On the Structure of the Tensor Square of the Natural Module of the Symmetric Group

2011 ◽  
Vol 18 (04) ◽  
pp. 589-610 ◽  
Author(s):  
Jürgen Müller ◽  
Johannes Orlob
Keyword(s):  
Symmetric Group ◽  
Prime Characteristic ◽  
Characteristic 2 ◽  
Natural Module ◽  
Tensor Square

We determine the submodule structure of the tensor square of the natural module of the symmetric group over a field of prime characteristic. We also determine the submodule structure of certain Young modules over a field of characteristic 2.

A modular version of Klyachko's theorem on Lie representations of the general linear group

2012 ◽  
Vol 153 (1) ◽  
pp. 79-98 ◽  
Author(s):  
R. M. BRYANT ◽  
MARIANNE JOHNSON
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Prime Characteristic ◽  
Tensor Power ◽  
Natural Module

AbstractKlyachko, in 1974, considered the tensor and Lie powers of the natural module for the general linear group over a field of characteristic 0 and showed that nearly all of the irreducible submodules of the rth tensor power also occur up to isomorphism as submodules of the rth Lie power. Here we prove an analogue for infinite fields of prime characteristic by showing, with some restrictions on r, that nearly all of the indecomposable direct summands of the rth tensor power also occur up to isomorphism as summands of the rth Lie power.

scholarly journals Higher symmetries of powers of the Laplacian and rings of differential operators

2017 ◽  
Vol 153 (4) ◽  
pp. 678-716 ◽  
Author(s):  
T. Levasseur ◽  
J. T. Stafford
Keyword(s):  
Lie Algebra ◽  
Differential Operators ◽  
Highest Weight ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Rings Of Differential Operators ◽  
Orthogonal Lie Algebra ◽  
Highest Weight Representation ◽  
Minimal Representations ◽  
Natural Module

We study the interplay between the minimal representations of the orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the algebra of symmetries$\mathscr{S}(\Box ^{r})$ of powers of the Laplacian $\Box$ on $\mathbb{C}^{n}$. The connection is made through the construction of a highest-weight representation of $\mathfrak{g}$ via the ring of differential operators ${\mathcal{D}}(X)$ on the singular scheme $X=(\mathtt{F}^{r}=0)\subset \mathbb{C}^{n}$, for $\mathtt{F}=\sum _{j=1}^{n}X_{i}^{2}\in \mathbb{C}[X_{1},\ldots ,X_{n}]$. In particular, we prove that $U(\mathfrak{g})/K_{r}\cong \mathscr{S}(\Box ^{r})\cong {\mathcal{D}}(X)$ for a certain primitive ideal $K_{r}$. Interestingly, if (and only if) $n$ is even with $r\geqslant n/2$, then both $\mathscr{S}(\Box ^{r})$ and its natural module ${\mathcal{A}}=\mathbb{C}[\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n},\ldots ,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n}]/(\Box ^{r})$ have a finite-dimensional factor. The same holds for the ${\mathcal{D}}(X)$-module ${\mathcal{O}}(X)$. We also study higher-dimensional analogues $M_{r}=\{x\in A:\Box ^{r}(x)=0\}$ of the module of harmonic elements in $A=\mathbb{C}[X_{1},\ldots ,X_{n}]$ and of the space of ‘harmonic densities’. In both cases we obtain a minimal $\mathfrak{g}$-representation that is closely related to the $\mathfrak{g}$-modules ${\mathcal{O}}(X)$ and ${\mathcal{A}}$. Essentially all these results have real analogues, with the Laplacian replaced by the d’Alembertian $\Box _{p}$ on the pseudo-Euclidean space $\mathbb{R}^{p,q}$ and with $\mathfrak{g}$ replaced by the real Lie algebra $\mathfrak{so}(p+1,q+1)$.

Recognising Tensor Products of Matrix Groups

1997 ◽  
Vol 07 (05) ◽  
pp. 541-559 ◽  
Author(s):  
C. R. Leedham-Green ◽  
E. A. O'Brien
Keyword(s):  
Tensor Product ◽  
Finite Fields ◽  
Tensor Products ◽  
Tensor Decomposition ◽  
Matrix Group ◽  
Matrix Groups ◽  
Natural Module

As a contribution to the project for recognising matrix groups defined over finite fields, we describe an algorithm for deciding whether or not the natural module for such a matrix group can be decomposed into a non-trivial tensor product. In the affirmative case, a tensor decomposition is returned. As one component, we develop algorithms to compute p-local subgroups of a matrix group.

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