The paper does not meet the standards of \"Food and Nutrition Sciences\".

This article has been retracted to straighten the academic record. In making this decision the Editorial Board follows COPE's Retraction Guidelines. The aim is to promote the circulation of scientific research by offering an ideal research publication platform with due consideration of internationally accepted standards on publication ethics. The Editorial Board would like to extend its sincere apologies for any inconvenience this retraction may have caused.

Editor guiding this retraction: Prof. Dr. Alessandra Bordoni (EiC of FNS)

The full retraction notice in PDF is preceding the original paper, which is marked \"RETRACTED\".

The paper does not meet the standards of \"Food and Nutrition Sciences\".

This article has been retracted to straighten the academic record. In making this decision the Editorial Board follows COPE's Retraction Guidelines. The aim is to promote the circulation of scientific research by offering an ideal research publication platform with due consideration of internationally accepted standards on publication ethics. The Editorial Board would like to extend its sincere apologies for any inconvenience this retraction may have caused.

Editor guiding this retraction: Prof. Dr. Alessandra Bordoni (EiC of FNS)

The full retraction notice in PDF is preceding the original paper, which is marked \"RETRACTED\".

Abstract:
In previous work, the authors established various bounds for the dimensions of degree $n$ cohomology and $\Ext$-groups, for irreducible modules of semisimple algebraic groups $G$ (in positive characteristic $p$) and (Lusztig) quantum groups $U_\zeta$ (at roots of unity $\zeta$). These bounds depend only on the root system, and not on the characteristic $p$ or the size of the root of unity $\zeta$. This paper investigates the rate of growth of these bounds. Both in the quantum and algebraic group situation, these rates of growth represent new and fundamental invariants attached to the root system $\Phi$. For quantum groups $U_\zeta$ with a fixed $\Phi$, we show the sequence $\{\max_{L\,{\text{irred}}}\dim \opH^n(U_\zeta,L)\}_n$ has polynomial growth independent of $\zeta$. In fact, we provide upper and lower bounds for the polynomial growth rate. Applications of these and related results for $\Ext^n_{U_\zeta}$ are given to Kazhdan-Lusztig polynomials. Polynomial growth in the algebraic group case remains an open question, though it is proved that $\{\log\max_{L\,\text{\rm irred}}\dim\opH^n(G,L)\}$ has polynomial growth $\leq 3$ for any fixed prime $p$ (and $\leq 4$ if $p$ is allowed to vary with $n$). We indicate the relevance of these issues to (additional structure for) the constants proposed in the theory of higher cohomology groups for finite simple groups with irreducible coefficients by Guralnick, Kantor, Kassabov, and Lubotzky \cite{GKKL}.

Abstract:
This paper considers Weyl modules for a simple, simply connected algebraic group over an algebraically closed field $k$ of positive characteristic $p\not=2$. The main result proves, if $p\geq 2h-2$ (where $h$ is the Coxeter number) and if the Lusztig character formula holds for all (irreducible modules with) regular restricted highest weights, then any Weyl module $\Delta(\lambda)$ has a $\Delta^p$-filtration, namely, a filtration with sections of the form $\Delta^p(\mu_0+p\mu_1):=L(\mu_0)\otimes\Delta(\mu_1)^{[1]}$, where $\mu_0$ is restricted and $\mu_1$ is arbitrary dominant. In case the highest weight $\lambda$ of the Weyl module $\Delta(\lambda)$ is $p$-regular, the $p$-filtration is compatible with the $G_1$-radical series of the module. The problem of showing that Weyl modules have $\Delta^p$-filtrations was first proposed as a worthwhile ("w\"unschenswert") problem in Jantzen's 1980 Crelle paper.

Abstract:
Let $G$ be a semisimple, simply connected algebraic group defined and split over a prime field ${\mathbb F}_p$ of positive characteristic. For a positive integer $r$, let $G_r$ be the $r$th Frobenius kernel of $G$. Let $Q$ be a projective indecomposable (rational) $G_r$-module. The well-known Humprheys-Verma conjecture (cf. \cite{Ballard}) asserts that the $G_r$-action on $Q$ lifts to an rational action of $G$ on $Q$. For $p\geq 2h-2$ (where $h$ is the Coxeter number of $G$), this conjecture was proved by Jantzen in 1980, improving on early work of Ballard. However, it remains open for general characteristics. In this paper, the authors establish several graded analogues of the Humphreys-Verma conjecture, valid for all $p$. The most general of our results, proved in full here, was announced (without proof) in an earlier paper. Another result relates the Humphreys-Verma conjecture to earlier work of Alperin, Collins, and Sibley on finite group representation theory. A key idea in all formulations involves the notion of a forced grading. The latter goes back, in particular, to the recent work of the authors, relating graded structures and $p$-filtrations. The authors anticipate that the Humphreys-Verma conjecture results here will lead to extensions to smaller characteristics of these earlier papers.

Abstract:
We investigate shock-wave solutions of the Einstein equations in the case when the speed of propagation is equal to the speed of light. The work extends the shock matching theory of Smoller and Temple, which characterizes solutions of the Einstein equations when the spacetime metric is only Lipschitz continuous across a hypersurface, to the lightlike case. After a brief introduction to general relativity, we develop the extension of a shock matching theory after introducing a previously known generalization of the second fundamental form by Barrabes and Israel. In the development of the theory we demonstrate that the matching of the generalized second fundamental form alone is not a sufficient condition for conservation conditions to hold across the interface. We then use this theory to construct a new exact solution of the Einstein equations that can be interpreted as an outgoing spherical shock wave that propagates at the speed of light. The solution is constructed by matching a Friedman Robertson Walker (FRW) metric, which is a geometric model for the universe, to a Tolman Oppenheimer Volkoff (TOV) metric, which models a static isothermal spacetime. The sound speeds, on each side of the shock, are constant and sub-luminous. Furthermore, the pressure and density are smaller at the leading edge of the shock, which is consistent with the Lax entropy condition in classical gas dynamics. However, the shock speed is greater than all the characteristic speeds. The solution also yields a surprising result in that the solution is not equal to the limit of previously known subluminous solutions as they tend to the speed of light.

Abstract:
Given a quasi-hereditary algebra $B$, we present conditions which guarantee that the algebra $\gr B$ obtained by grading $B$ by its radical filtration is Koszul and at the same time inherits the quasi-hereditary property and other good Lie-theoretic properties that $B$ might possess. The method involves working with a pair $(A,{\mathfrak a})$ consisting of a quasi-hereditary algebra $A$ and a (positively) graded subalgebra $\mathfrak a$. The algebra $B$ arises as a quotient $B=A/J$ of $A$ by a defining ideal $J$ of $A$. Along the way, we also show that the standard (Weyl) modules for $B$ have a structure as graded modules for $\mathfrak a$. These results are applied to obtain new information about the finite dimensional algebras (e.g., the $q$-Schur algebras) which arise as quotients of quantum enveloping algebras. Further applications, perhaps the most penetrating, yield results for the finite dimensional algebras associated to semisimple algebraic groups in positive characteristic $p$. These results require, at least presently, considerable restrictions on the size of $p$.

Abstract:
This paper has two parts. The main goal, carried out in Part I, is to survey some recent work by the authors in which "forced" grading constructions have played a significant role in the representation theory of semisimple algebraic groups $G$ in positive characteristic. The constructions begin with natural finite dimensional quotients of the distribution algebras Dist$(G)$, but then "force" gradings into the picture by passing to positively graded algebras constructed from ideal filtrations of these quotients. This process first guaranteed a place for itself by proving, for large primes, that all Weyl modules have $p$-Weyl filtrations. Later it led, under similar circumstances, to a new "good filtration" result for restricted Lie algebra Ext groups between restricted irreducible $G$-modules. In the process of proving these results, a new kind of graded algebra was invented, called a Q-Koszul algebra. Recent conjectures suggest these algebras arise in forced grading constructions as above, from quotients of Dist$(G)$, even for small primes and even in settings possibly involving singular weights. Related conjectures suggest a promising future for using Kazhdan-Lusztig theory to relate quantum and algebraic group cohomology and Ext groups in these same small prime and possibly singular weight settings. A part of one of these conjectures is proved in Part II of this paper. The proof is introduced by remarks of general interest on positively graded algebras and Morita equivalence, followed by a discussion of recent Koszulity results of Shan-Varagnalo-Vasserot, observing some extensions.

Abstract:
Let $N$ be a normal subgroup of a group $G$. An $N$-module $Q$ is $G$-stable provided that $Q$ is equivalent to the twist $Q^g$ of $Q$ by $g$, for every $g\in G$. If the action of $N$ on $Q$ extends to an action of $G$ on $Q$, $Q$ is obviously $G$-stable, but the converse need not hold. A famous conjecture in the modular representation theory of reductive algebraic groups $G$ asserts that the (obviously $G$-stable) projective indecomposable modules (PIMs) $Q$ for the Frobenius kernels of $G$ have a $G$-module structure. It is sometimes just as useful (for a general module $Q$) to know that a finite direct sum $Q^{\oplus n}$ of $Q$ has a compatible $G$-module structure. In this paper, this property is called numerical stability. In recent work (arXiv:0909.5207v2), the authors established numerical stability in the special case of PIMs. We provide in this paper a more general context for that result, working in the context of group schemes and a suitable version of $G$-stability, called strong $G$-stability. Among our results here is the presentation of a homological obstruction to the existence of a $G$-module structure, on strongly $G$-stable modules, and a tensor product approach to killing the obstruction.