6,554 research outputs found

### Finite dimensional algebras and cellular systems

We introduce the notion of a cellular system in order to deal with
quasi-hereditary algebras. We shall prove that a necessary and sufficient
condition for an algebra to be quasi-hereditary is the existence of a full
divisible cellular system. As a further application, we prove that the
existence of a full local cellular system is a sufficient condition for a
standardly stratified algebra.Comment: 15 page

### Frobenius morphisms and representations of algebras

By introducing Frobenius morphisms $F$ on algebras $A$ and their modules over
the algebraic closure {{\bar \BF}}_q of the finite field \BF_q of $q$
elements, we establish a relation between the representation theory of $A$ over
{{\bar \BF}}_q and that of the $F$-fixed point algebra $A^F$ over \BF_q.
More precisely, we prove that the category \modh A^F of finite dimensional
$A^F$-modules is equivalent to the subcategory of finite dimensional $F$-stable
$A$-modules, and, when $A$ is finite dimensional, we establish a bijection
between the isoclasses of indecomposable $A^F$-modules and the $F$-orbits of
the isoclasses of indecomposable $A$-modules. Applying the theory to
representations of quivers with automorphisms, we show that representations of
a modulated quiver (or a species) over \BF_q can be interpreted as $F$-stable
representations of a corresponding quiver over {{\bar \BF}}_q. We further
prove that every finite dimensional hereditary algebra over \BF_q is Morita
equivalent to some $A^F$, where $A$ is the path algebra of a quiver $Q$ over
{{\bar \BF}}_q and $F$ is induced from a certain automorphism of $Q$. A close
relation between the Auslander-Reiten theories for $A$ and $A^F$ is
established. In particular, we prove that the Auslander-Reiten (modulated)
quiver of $A^F$ is obtained by "folding" the Auslander-Reiten quiver of $A$.
Finally, by taking Frobenius fixed points, we are able to count the number of
indecomposable representations of a modulated quiver with a given dimension
vector and to establish part of Kac's theorem for all finite dimensional
hereditary algebras over a finite field.Comment: 28 page

### Small Representations for Affine q-Schur Algebras

When the parameter $q\in\mathbb C^*$ is not a root of unity, simple modules
of affine $q$-Schur algebras have been classified in terms of Frenkel--Mukhin's
dominant Drinfeld polynomials (\cite[4.6.8]{DDF}). We compute these Drinfeld
polynomials associated with the simple modules of an affine $q$-Schur algebra
which come from the simple modules of the corresponding $q$-Schur algebra via
the evaluation maps.Comment: 21 Page

### Modified FEA and ExtraTree algorithm for transformer Green's function modelling

The Green's function of a transformer is essential for prediction of its
vibration. As the Green's function cannot be measured directly and completely,
the finite element analysis (FEA) is typically used for its estimation.
However, because of the complexity of the transformer structure, the
calculations involved in FEA are time consuming. Therefore, in this paper, a
method based on FEA modified by an extremely random tree algorithm call
ExtraTree is proposed to efficiently estimate the Green's function of a
transformer. A subset of the frequency response functions from FEA will be
selected by a genetic algorithm that can well present the structural variation.
The FEA calculation time can be reduced by simply calculating the frequency
response functions on this subset and predicting remainder using the trained
ExtraTree model. The errors introduced in this method can be estimated from the
corresponding frequency and the genetic algorithm error.Comment: 8 pages, 5 figures, 3 tables, submitting to internoise 201

### On bases of quantized enveloping algebras

We give a systematic description of many monomial bases for a given quantized
enveloping algebra and of many integral monomial bases for the associated
Lusztig $\mathbb Z[v,v^{-1}]$-form. The relations between monomial bases, PBW
bases and canonical bases are also discussed.Comment: 13 page

### Standard multipartitions and a combinatorial affine Schur-Weyl duality

We introduce the notion of standard multipartitions and establish a
one-to-one correspondence between standard multipartitions and irreducible
representations with integral weights for the affine Hecke algebra of type A
with a parameter q which is not a root of unity. We then extend the
correspondence to all Kleshchev multipartitions for Ariki-Koike algebras of
integral type. By the affine Schur--Weyl duality, we further extend this to a
correspondence between standard multipartitions and Drinfeld multipolynomials
of integral type whose associated irreducible polynomial representations
completely determine all irreducible polynomial representations for the quantum
loop algebra. We will see, in particular, the notion of standard
multipartitions gives rise to a combinatorial description of the affine
Schur--Weyl duality in terms of a column-reading vs. row reading of residues of
a multipartition.Comment: 24 page

### Monomial bases for quantum affine sl_n

We use the idea of generic extensions to investigate the correspondence
between the isomorphism classes of nilpotent representations of a cyclic quiver
and the orbits in the corresponding representation varieties. We endow the set
$\cal M$ of such isoclasses with a monoid structure and identify the submonoid
$\cal M_c$ generated by simple modules. On the other hand, we use the partial
ordering on the orbits (i.e., the Bruhat-Chevalley type ordering) to induce a
poset structure on $\cal M$ and describe the poset ideals generated by an
element of the submonoid $\cal M_c$ in terms of the existence of a certain
composition series of the corresponding module. As applications of these
results, we generalize some results of Ringel involving special words to
results with no restriction on words and obtain a systematic description of
many monomial bases for any given quantum affine ${\frak {sl}}_n$.Comment: 24 page

### Ariki-Koike Algebras with Semisimple Bottoms

We investigated the representation thoery of an Ariki-Koike algebra whose
Poincare polynomial associated with the "bottom", i.e., the subgroup on which
the symmetric group acts, is non-zero in the base field. We proved that the
module category of such an Ariki-Koike algebra is Morita equivalent to the
module category of a direct sum of tensor products of Hecke algebras associated
with certain symmetric groups. We also generalized this Morita equivalence
theorem to give a Morita equivalenve between a $q$-Schur$^m$ algebra and a
direct sum of tensor products of certain $q$-Schur algebras.Comment: 20 pages. Math. Zeit. (to appear

### Canonical bases for the quantum linear supergroups

We give a combinatorial construction for the canonical bases of the
$\pm$-parts of the quantum enveloping superalgebra \bfU(\mathfrak{gl}_{m|n})
and discuss their relationship with the Kazhdan-Lusztig bases for the quantum
Schur superalgebras \bsS(m|n,r) introduced in \cite{DR}. We will also extend
this relationship to the induced bases for simple polynomial representations of
\bfU(\mathfrak{gl}_{m|n}).Comment: 30 page

### The queer q-Schur superalgebra

As a natural generalization quantum Schur algebras associated with the Hecke
algebra of the symmetric group, we introduce the quantum Schur superalgebra of
type Q associated with the Hecke-Clifford superalgebra, which, by definition,
is the endomorphism algebra of the induced module over the Hecke-Clifford
superalgebra from certain permutation modules over the Hecke algebra of the
symmetric group. We will describe certain integral bases for these
superalgebras in terms of matrices and will establish the base change property
for them. We will also identify the quantum Schur superalgebra of type Q with
the quantum queer Schur superalgebras investigated in the context of quantum
queer supergroups and then provide a classification of their irreducible
representations over a certain extension of the field of complex rational
functions.Comment: 27 pages, version 2 with an appendix added, to appear in Journal of
the Australian Mathematical Society, available at
https://doi.org/10.1017/S144678871700033

- β¦