The ladder crystal

n this paper I introduce a new description of the crystal $B(\Lambda_0)$ of $\hat{\mathfrak{sl}_\ell}$. As in the Misra-Miwa model of $B(\Lambda_0)$, the nodes of this crystal are indexed by partitions and the $i$-arrows correspond to adding a box of residue $i$. I then show that the two models are equivalent by interpreting the operation of regularization introduced by James as a crystal isomorphism

The biggest vested interest of all: how government lobbies to restrict individual rights and freedom

The Federal Treasurer Wayne Swan wrote in The Monthly in March 2012 that : Australia’s fair go is today under threat from a new source."To be blunt, the rising power of vested interests is undermining our equality and threatening our democracy."But not all vested interests are private corporations. This paper draws attention to two statutory agencies of the Commonwealth Government that have an explicit, legislatively - defined function to lobby and advocate for public policy change – the Australian National Preventive Health Agency and the Australian Human Rights Commission.These two agencies are effectively taxpayer funded lobbyists, embedded in the public policy process, enjoying privileged access to the institutions of government.The Australian National Preventative Health Agency (ANPHA) received $57,718,000 in the 2012 - 13 Federal Budget to “driv[e] the national capacity for change and innovation around preventive health policies and programs.”ANPHA publicly advocates and privately lobbies for a wide range of Nanny State restrictions on alcohol, tobacco, and unhealthy food.The Australian Human Rights Commission (AHRC) received$23,133,000 in the 2012 - 13 Federal Budget . One oF its primary tasks is to “promote an understanding and acceptance ... of human rights in Australia ... undertake research and educational programs” and “develop laws, policies and programs” for parliament to enact. ( Unfortunately, the AHRC does not disclose how much it of it s budget it directs towards this task.)However the human rights that the AHRC chooses to promote and advocate are highly selective, favouring certain rights above others.As well as being policy lobbyists in their own right, AHRC and ANPHA are central to a pattern of relationships between the government and non - government sectors. Taxpayer money is being used to lobby for the allocation of more taxpayer money.One - third of the submissions to the Preventative Health Taskforce – which established the Australian National Preventive Health Agency – were from bodies which received large amounts of taxpayer funding

The down operator and expansions of near rectangular k-Schur functions

We prove that the Lam-Shimozono "down operator" on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of "near rectangles" in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood--Richardson coefficients

Noncommutativity as a colimit

Every partial algebra is the colimit of its total subalgebras. We prove this result for partial Boolean algebras (including orthomodular lattices) and the new notion of partial C*-algebras (including noncommutative C*-algebras), and variations such as partial complete Boolean algebras and partial AW*-algebras. The first two results are related by taking projections. As corollaries we find extensions of Stone duality and Gelfand duality. Finally, we investigate the extent to which the Bohrification construction, that works on partial C*-algebras, is functorial.Comment: 22 pages; updated theorem 15, added propoisition 3

Expansion of kk-Schur functions for maximal kk-rectangles within the affine nilCoxeter algebra

We give several explicit combinatorial formulas for the expansion of k-Schur functions indexed by maximal rectangles in terms of the standard basis of the affine nilCoxeter algebra. Using our result, we also show a commutation relation of k-Schur functions corresponding to rectangles with the generators of the affine nilCoxeter algebra.Comment: to appear in Journal of Combinatorics, 28 page

Combinatorics of (ℓ,0)(\ell,0)-JM partitions, ℓ\ell-cores, the ladder crystal and the finite Hecke algebra

The following thesis contains results on the combinatorial representation theory of the finite Hecke algebra $H_n(q)$. In Chapter 2 simple combinatorial descriptions are given which determine when a Specht module corresponding to a partition $\lambda$ is irreducible. This is done by extending the results of James and Mathas. These descriptions depend on the crystal of the basic representation of the affine Lie algebra $\widehat{\mathfrak{sl}_\ell}$. In Chapter 3 these results are extended to determine which irreducible modules have a realization as a Specht module. To do this, a new condition of irreducibility due to Fayers is combined with a new description of the crystal from Chapter 2. In Chapter 4 a bijection of cores first described by myself and Monica Vazirani is studied in more depth. Various descriptions of it are given, relating to the quotient $\widetilde{S_\ell}/{S_\ell}$ and to the bijection given by Lapointe and Morse