Realization of affine type A Kirillov-Reshetikhin crystals via polytopes

On the polytope defined in Feigin, Fourier, and Littelmann (2011), associated to any rectangle highest weight, we define a structure of an type $A_n$-crystal. We show, by using the Stembridge axioms, that this crystal is isomorphic to the one obtained from Kashiwara's crystal bases theory. Further we define on this polytope a bijective map and show that this map satisfies the properties of a weak promotion operator. This implies in particular that we provide an explicit realization of Kirillov-Reshetikhin crystals for the affine type $A^{(1)}_n$ via polytopes

Consumption and Social Welfare Politics: The Effect of Credit and China

Analyzing data from 20 OECD countries over the period of 1995-2007, the present article investigates whether the factors that contributed to households’ consumption opportunities have had any impact on the way governments in advanced societies respond to income inequalities. In addressing this question, the article particularly focuses on access to credit, and low-wage imports, from China in particular, as two mechanisms that have contributed to an increase in household consumption opportunities. The results show a highly significant inverse relation between these two factors and social welfare effort. As imports from China and availability of credit increase, the social welfare effort seems to decrease. These findings prompt us to think beyond the established arguments about progressive politics in the neoliberal era. The article also contributes to the burgeoning literature on the political and social implications of credit expansion, and of the rise of China in world trade.

PBW degenerations of Lie superalgebras and their typical representations

We introduce the PBW degeneration for basic classical Lie superalgebras and construct for all type I, $\mathfrak{osp}(1,2n)$ and exceptional Lie superalgebras new monomial bases. These bases are parametrized by lattice points in convex lattice polytopes, sharing useful properties such as the integer decomposition property. This paper is the first step towards extending the framework of PBW degenerations to the Lie superalgebra setting

Twisted Demazure modules, fusion product decomposition and twisted Q--systems

In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the twisted current algebras. These modules are indexed by an $|R^+|$-tuple of partitions \bxi=(\xi^{\alpha})_{\alpha\in R^+} satisfying a natural compatibility condition. We give three equivalent presentations of these modules and show that for a particular choice of \bxi these modules become isomorphic to Demazure modules in various levels for the twisted affine algebras. As a consequence we see that the defining relations of twisted Demazure modules can be greatly simplified. Furthermore, we investigate the notion of fusion products for twisted modules, first defined in \cite{FL99} for untwisted modules, and use the simplified presentation to prove a fusion product decomposition of twisted Demazure modules. As a consequence we prove that twisted Demazure modules can be obtained by taking the associated graded modules of (untwisted) Demazure modules for simply-laced affine algebras. Furthermore we give a semi-infinite fusion product construction for the irreducible representations of twisted affine algebras. Finally, we prove that the twisted $Q$-sytem defined in \cite{HKOTT02} extends to a non-canonical short exact sequence of fusion products of twisted Demazure modules

A combinatorial formula for graded multiplicities in excellent filtrations

A filtration of a representation whose successive quotients are isomorphic to Demazure modules is called an excellent filtration. In this paper we study graded multiplicities in excellent filtrations of fusion products for the current algebra $\mathfrak{sl}_2[t]$. We give a combinatorial formula for the polynomials encoding these multiplicities in terms of two dimensional lattice paths. Corollaries to our main theorem include a combinatorial interpretation of various objects such as the coeffficients of Ramanujan's fifth order mock theta functions $\phi_0, \phi_1, \psi_0, \psi_1$, Kostka polynomials for hook partitions and quotients of Chebyshev polynomials. We also get a combinatorial interpretation of the graded multiplicities in a level one flag of a local Weyl module associated to the simple Lie algebras of type $B_n \text{ and } G_2$

Borel-de Siebenthal theory for affine reflection systems

We develop a Borel-de Siebenthal theory for affine reflection systems by classifying their maximal closed subroot systems. Affine reflection systems (introduced by Loos and Neher) provide a unifying framework for root systems of finite-dimensional semi-simple Lie algebras, affine and toroidal Lie algebras, and extended affine Lie algebras. In the special case of nullity $k$ toroidal Lie algebras, we obtain a one-to-one correspondence between maximal closed subroot systems with full gradient and triples $(q,(b_i),H)$, where $q$ is a prime number, $(b_i)$ is a $n$-tuple of integers in the interval $[0,q-1]$ and $H$ is a $(k\times k)$ Hermite normal form matrix with determinant $q$. This generalizes the $k=1$ result of Dyer and Lehrer in the setting of affine Lie algebras