Aug 2026· Algebras and Representation Theory· 0 citations· 7 references
Abstract
<jats:p>
Let
<jats:italic>p</jats:italic>
be an odd prime. Let
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$${G = SL_2(\mathbb {F}_p)}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>G</mml:mi>
<mml:mo>=</mml:mo>
<mml:mi>S</mml:mi>
<mml:msub>
<mml:mi>L</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:msub>
<mml:mi>F</mml:mi>
<mml:mi>p</mml:mi>
</mml:msub>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
and let
<jats:italic>B</jats:italic>
denote the subgroup of upper triangular matrices of
<jats:italic>G</jats:italic>
. Finally, let
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$${\mathbb {F}}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>F</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
be an algebraically closed field of characteristic
<jats:italic>p</jats:italic>
. The Green correspondence gives a bijection between the non-projective indecomposable
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$${\mathbb {F}[G]}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mo>[</mml:mo>
<mml:mi>G</mml:mi>
<mml:mo>]</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
modules and non-projective indecomposable
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$${\mathbb {F}[B]}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mo>[</mml:mo>
<mml:mi>B</mml:mi>
<mml:mo>]</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
modules, realised by restriction and induction. In this paper, after recalling a suitable description of the non-projective indecomposable modules for these group algebras, we explicitly describe the Green correspondence bijection. We do this by pinpointing the modules’ position on the Stable Auslanden-Reiten quivers. Finally, we obtain two corollaries in terms of this description: formula for lifting the
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$${\mathbb {F}[B]}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mo>[</mml:mo>
<mml:mi>B</mml:mi>
<mml:mo>]</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
module decomposition of an
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$${\mathbb {F}[G]}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>F</mml:mi>
<mml:mo>[</mml:mo>
<mml:mi>G</mml:mi>
<mml:mo>]</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
module, and a complete description of
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$${\text { Ind}_B^G}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mspace/>
<mml:msubsup>
<mml:mtext>Ind</mml:mtext>
<mml:mi>B</mml:mi>
<mml:mi>G</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
and
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$${\text { Res}^G_B}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mspace/>
<mml:msubsup>
<mml:mtext>Res</mml:mtext>
<mml:mi>B</mml:mi>
<mml:mi>G</mml:mi>
</mml:msubsup>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
.
</jats:p>
<jats:p>
In the first part of the paper we show that every closed subspace of
<jats:italic>JT</jats:italic>
or
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$JT^*$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>J</mml:mi>
<mml:msup>
<mml:mi>T</mml:mi>
<mml:mo>∗</mml:mo>
</mml:msup>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
contains
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\ell _2$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>ℓ</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
complemented in
<jats:italic>JT</jats:italic>
or
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$JT^*$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>J</mml:mi>
<mml:msup>
<mml:mi>T</mml:mi>
<mml:mo>∗</mml:mo>
</mml:msup>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
respectively, and
<jats:italic>JT</jats:italic>
contains uncomplemented copies of
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\ell _2$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>ℓ</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
. As a result, the predual
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {B}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>B</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
of
<jats:italic>JT</jats:italic>
, as well as the spaces
<jats:italic>JT</jats:italic>
and
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$JT^*$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>J</mml:mi>
<mml:msup>
<mml:mi>T</mml:mi>
<mml:mo>∗</mml:mo>
</mml:msup>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, are subprojective and superprojective. In the second part, we prove that every weakly Cauchy sequence that is not weakly convergent in
<jats:italic>JT</jats:italic>
has a subsequence equivalent to the basis of
<jats:italic>J</jats:italic>
. Hence, every non-reflexive subspace of
<jats:italic>JT</jats:italic>
contains an isomorphic copy of
<jats:italic>J</jats:italic>
, and every Schauder basic sequence in
<jats:italic>JT</jats:italic>
has a subsequence which is equivalent either to the basis of
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\ell _2$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>ℓ</mml:mi>
<mml:mn>2</mml:mn>
</mml:msub>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
or to the basis of
<jats:italic>J</jats:italic>
. Moreover these subspaces may be selected to be complemented in
<jats:italic>JT</jats:italic>
.
</jats:p>
S. Argyros, Manuel González, P. Motakis· Revista Matemática Compluten...· 0 citations
<jats:p>
We provide an equivariant extension of Carlsson’s BGG correspondence in characteristic two. As an application we classify perfect cochain complexes of
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$(\mathbb {Z}/2\times \mathbb {Z}/2)\rtimes Q$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>Z</mml:mi>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>×</mml:mo>
<mml:mi>Z</mml:mi>
<mml:mo>/</mml:mo>
<mml:mn>2</mml:mn>
<mml:mo>)</mml:mo>
<mml:mo>⋊</mml:mo>
<mml:mi>Q</mml:mi>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-representations with four-dimensional total homology for finite groups
<jats:italic>Q</jats:italic>
of odd order. We deduce that cochain complexes of finite, free
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$A_4$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>A</mml:mi>
<mml:mn>4</mml:mn>
</mml:msub>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-CW complexes with four-dimensional total homology are rigid: They are determined by the degrees of the nonzero homology groups.
</jats:p>
Henrik Rüping, Marc Stephan· Algebras and Representation...· 0 citations
<jats:p>
In this paper, we introduce the notion of generalized
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {W}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>W</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-Gorenstein modules respect to some subclass
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {W}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>W</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, extending the classical notion of Gorenstein projective modules. By exploiting the correspondence between projective modules over the endomorphism ring
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\textrm{End}_R(C)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mtext>End</mml:mtext>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>C</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
of a module
<jats:italic>C</jats:italic>
and elements of its additive closure
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {W}=\textrm{Add}_R(C)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>W</mml:mi>
<mml:mo>=</mml:mo>
<mml:msub>
<mml:mtext>Add</mml:mtext>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>C</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, we establish a fundamental correspondence between Gorenstein projective
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\textrm{End}_R(C)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mtext>End</mml:mtext>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>C</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-modules and generalized
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\textrm{Add}_R(C)$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:msub>
<mml:mtext>Add</mml:mtext>
<mml:mi>R</mml:mi>
</mml:msub>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>C</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-Gorenstein modules. This result refines existing relative homological settings and provides a natural extension of well-known results in Gorenstein homological algebra. We explore key properties, such as closure under direct summands and sums, and identify conditions under which the class of generalized
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\mathcal {W}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>W</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-Gorenstein modules coincides with other classes of modules, like Gorenstein projective modules.
</jats:p>
Driss Bennis, C. Lomp, Abderrazak Nassir· Algebras and Representation...· 0 citations
<jats:p>
We provide a geometric model for the free
<jats:italic>X</jats:italic>
-generated
<jats:italic>F</jats:italic>
-restriction semigroup in the extended signature
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$${(\cdot , ^+,^{\mathfrak {m}},\lambda )}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mo>·</mml:mo>
<mml:msup>
<mml:mo>,</mml:mo>
<mml:mo>+</mml:mo>
</mml:msup>
<mml:msup>
<mml:mo>,</mml:mo>
<mml:mi>m</mml:mi>
</mml:msup>
<mml:mo>,</mml:mo>
<mml:mi>λ</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, where the unary operation
<jats:sup>m</jats:sup>
maps an element
<jats:italic>a</jats:italic>
to the maximum element
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$a^{\mathfrak {m}}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>a</mml:mi>
<mml:mi>m</mml:mi>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
of its
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\sigma $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>σ</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-class, and the constant
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\lambda $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>λ</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
is the unique left identity. This model is based on a certain quotient of the Cayley graph expansion of the free monoid
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$X^*$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>X</mml:mi>
<mml:mo>∗</mml:mo>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
with respect to the extended set of generators
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$X\cup \overline{X^*}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>X</mml:mi>
<mml:mo>∪</mml:mo>
<mml:mover>
<mml:msup>
<mml:mi>X</mml:mi>
<mml:mo>∗</mml:mo>
</mml:msup>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:mrow>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
, where the generators from
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\overline{X^*}$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mover>
<mml:msup>
<mml:mi>X</mml:mi>
<mml:mo>∗</mml:mo>
</mml:msup>
<mml:mo>¯</mml:mo>
</mml:mover>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
are in a bijection with the free monoid
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$X^*$$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msup>
<mml:mi>X</mml:mi>
<mml:mo>∗</mml:mo>
</mml:msup>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
and serve to capture the maximum elements of
<jats:inline-formula>
<jats:alternatives>
<jats:tex-math>$$\sigma $$</jats:tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>σ</mml:mi>
</mml:math>
</jats:alternatives>
</jats:inline-formula>
-classes of the quotient. We also provide models for the free
<jats:italic>X</jats:italic>
-generated strong and perfect
<jats:italic>F</jats:italic>
-restriction semigroups in the same extended signature. The constructed models enable us to solve the word problems for all the free objects under consideration.
</jats:p>