On the characters of the Sylow p-subgroups of untwisted Chevalley groups Y_n(p^a)

Let $UY_n(q)$ be a Sylow p-subgroup of an untwisted Chevalley group $Y_n(q)$ of rank n defined over $\mathbb{F}_q$ where q is a power of a prime p. We partition the set $Irr(UY_n(q))$ of irreducible characters of $UY_n(q)$ into families indexed by antichains of positive roots of the root system of type $Y_n$. We focus our attention on the families of characters of $UY_n(q)$ which are indexed by antichains of length 1. Then for each positive root $\alpha$ we establish a one to one correspondence between the minimal degree members of the family indexed by $\alpha$ and the linear characters of a certain subquotient $\overline{T}_\alpha$ of $UY_n(q)$. For $Y_n = A_n$ our single root character construction recovers amongst other things the elementary supercharacters of these groups. Most importantly though this paper lays the groundwork for our classification of the elements of $Irr(UE_i(q))$, $6 \le i \le 8$ and $Irr(UF_4(q))$

On the characters of Sylow pp-subgroups of finite Chevalley groups G(pf)G(p^f) for arbitrary primes

We develop in this work a method to parametrize the set $\mathrm{Irr}(U)$ of irreducible characters of a Sylow $p$-subgroup $U$ of a finite Chevalley group $G(p^f)$ which is valid for arbitrary primes $p$, in particular when $p$ is a very bad prime for $G$. As an application, we parametrize $\mathrm{Irr}(U)$ when $G=\mathrm{F}_4(2^f)$.Comment: 22 page

Avalanche dynamics of elastic interfaces

Slowly driven elastic interfaces, such as domain walls in dirty magnets, contact lines, or cracks proceed via intermittent motion, called avalanches. We develop a field-theoretic treatment to calculate, from first principles, the space-time statistics of instantaneous velocities within an avalanche. For elastic interfaces at (or above) their (internal) upper critical dimension d >= d_uc (d_uc = 2, 4 respectively for long-ranged and short-ranged elasticity) we show that the field theory for the center of mass reduces to the motion of a point particle in a random-force landscape, which is itself a random walk (ABBM model). Furthermore, the full spatial dependence of the velocity correlations is described by the Brownian-force model (BFM) where each point of the interface sees an independent Brownian-force landscape. Both ABBM and BFM can be solved exactly in any dimension d (for monotonous driving) by summing tree graphs, equivalent to solving a (non-linear) instanton equation. This tree approximation is the mean-field theory (MFT) for realistic interfaces in short-ranged disorder. Both for the center of mass, and for a given Fourier mode q, we obtain probability distribution functions (PDF's) of the velocity, as well as the avalanche shape and its fluctuations (second shape). Within MFT we find that velocity correlations at non-zero q are asymmetric under time reversal. Next we calculate, beyond MFT, i.e. including loop corrections, the 1-time PDF of the center-of-mass velocity du/dt for dimension d< d_uc. The singularity at small velocity P(du/dt) ~ 1/(du/dt)^a is substantially reduced from a=1 (MFT) to a = 1 - 2/9 (4-d) + ... (short-ranged elasticity) and a = 1 - 4/9 (2-d) + ... (long-ranged elasticity). We show how the dynamical theory recovers the avalanche-size distribution, and how the instanton relates to the response to an infinitesimal step in the force.Comment: 68 pages, 72 figure

Functional Renormalization for Disordered Systems, Basic Recipes and Gourmet Dishes

We give a pedagogical introduction into the functional renormalization group treatment of disordered systems. After a review of its phenomenology, we show why in the context of disordered systems a functional renormalization group treatment is necessary, contrary to pure systems, where renormalization of a single coupling constant is sufficient. This leads to a disorder distribution, which after a finite renormalization becomes non-analytic, thus overcoming the predictions of the seemingly exact dimensional reduction. We discuss, how the non-analyticity can be measured in a simulation or experiment. We then construct a renormalizable field theory beyond leading order. We discuss an elastic manifold embedded in N dimensions, and give the exact solution for N to infinity. This is compared to predictions of the Gaussian replica variational ansatz, using replica symmetry breaking. We further consider random field magnets, and supersymmetry. We finally discuss depinning, both isotropic and anisotropic, and universal scaling function.Comment: 29 page

Field Theory of Disordered Elastic Interfaces at 3-Loop Order: The β\beta-Function

We calculate the effective action for disordered elastic manifolds in the ground state (equilibrium) up to 3-loop order. This yields the renormalization-group $\beta$-function to third order in $\epsilon=4-d$, in an expansion in the dimension $d$ around the upper critical dimension $d=4$. The calculations are performed using exact RG, and several other techniques, which allow us to resolve consistently the problems associated with the cusp of the renormalized disorder.Comment: This is the first part of arXiv:1707.09802v1. The remaining part is in arXiv:1707.09802v2. 47 pages, 67 figures. v2: typos corrected and hyper-ref enable