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Together with David Schlang we computed the discriminants of the invariant Hermitian forms for all indicator $o$ even degree absolutely irreducible characters of the ATLAS groups supplementing the tables of orthogonal determinants computed in collaboration with Richard Parker, Tobias Braun and Thomas Breuer. The methods that are used in the unitary case are described in this paper. A character has a well defined unitary discriminant, if and only if it is unitary stable, i.e. all irreducible unitary constituents have even degree. Computations for large degree characters are only possible because of a new method called {\em unitary condensation}. A suitable automorphism helps to single out a square class of the real subfield of the character field consisting of representatives of the discriminant of the invariant Hermitian forms. This square class can then be determined modulo enough primes.

The paper develops elementary linear algebra methods to compute the determinants of the tensor symmetrizations of quadratic and hermitian forms over fields of good characteristic. Explicit results are given for the partitions $(n)$, $(1^n)$, $(2,1^{n-2})$ and $(3,1^{n-3})$ as well as for all partitions of $n\leq 7$. For orthogonal groups these symmetrizations are not irreducible and we continue to find the determinants of their irreducible constituents, the refined symmetrizations, over fields of characteristic 0.

We give a full list of the orthogonal determinants of the even degree indicator '+' ordinary irreducible characters of $\mathrm{SL}_3(q)$ and $\mathrm{SU}_3(q)$.

Let $K$ be a field, $\Gamma $ a finite group of field automorphisms of $K$, $F$ the $\Gamma $-fixed field in $K$ and $G\leq $GL$_v(K)$ a finite matrix group. Then the action of $\Gamma $ defines a grading on the symmetric algebra of the $F$-space $K^v$ which we use to introduce the notion of homogeneous $\Gamma $-conjugate invariants of $G$. We apply this new grading in invariant theory to broaden the connection between codes and invariant theory by introducing $\Gamma $-conjugate complete weight enumerators of codes. The main result of this paper applies the theory from Nebe, Rains, Sloane to show that under certain extra conditions these new weight enumerators generate the ring of $\Gamma $-conjugate invariants of the associated Clifford-Weil groups. As an immediate consequence we obtain a result by Bannai etal that the complex conjugate weight enumerators generate the ring of complex conjugate invariants of the complex Clifford group. Also the Schur-Weyl duality conjectured and partly shown by Gross etal can be derived from our main result.

The linear spaces that are fixed by a given nilpotent $n \times n$ matrix form a subvariety of the Grassmannian. We classify these varieties for small $n$. Mutiah, Weekes and Yacobi conjectured that their radical ideals are generated by certain linear forms known as shuffle equations. We prove this conjecture for $n \leq 7$, and we disprove it for $n=8$. The question remains open for nilpotent matrices arising from the affine Grassmannian., Comment: 14 pages, some updates based on referee comments

An ordinary character $\chi $ of a finite group is called orthogonally stable, if all non-degenerate invariant quadratic forms on any module affording the character $\chi $ have the same discriminant. This is the orthogonal discriminant, $\disc(\chi )$, of $\chi $, a square class of the character field. Based on experimental evidence we conjecture that the orthogonal discriminant is always an odd square class in the sense of Definition 1.4. This note proves this conjecture for finite solvable groups. For $p$-group there is an explicit formula for $\disc(\chi )$ that reads $\disc(\chi ) = (-p)^{\chi(1)/2}$ if $p\equiv 3 \pmod{4}$ and $\disc (\chi ) = (-1)^{\chi(1)/2}$ for $p=2$.

A character (ordinary or modular) is called orthogonally stable if all non-degenerate quadratic forms fixed by representations with those constituents have the same determinant mod squares. We show that this is the case provided there are no odd-degree orthogonal constituents. We further show that if the reduction mod p of an ordinary character is orthogonally stable, this determinant is the reduction mod p of the ordinary one. In particular, if the characteristic does not divide the group order, we immediately see in which orthogonal group it lies. We sketch methods for computing this determinant, and give some examples.

Let $G$ be a finite group and $K$ a finite field of characteristic $2$. Denote by $t$ the $2$-rank of the commutator factor group $G/G'$ and by $s$ the number of self-dual simple $KG$-modules. Then the Witt group of equivariant quadratic forms $\WQ (K,G)$ is isomorphic to an elementary abelian $2$-group of rank $s+t$.

Bolytropes are bounded subsets of an affine building that consist of all points that have distance at most $r$ from some polytrope. We prove that the points of a bolytrope describe the set of all invariant lattices of a bolytrope order, generalizing the correspondence between polytropes and graduated orders., Comment: Minor revisions according to referee's suggestions. To appear in International Journal of Number Theory

For an irreducible orthogonal character $\chi $ of even degree there is a unique square class $\det({\chi })$ in the character field such that the invariant quadratic forms in any $L$-representation affording $\chi $ have determinant in $\det({\chi })(L^*)^2$., Comment: Added a new Section 2.2

We apply tropical geometry to study matrix algebras over a field with valuation. Using the shapes of min-max convexity, known as polytropes, we revisit the graduated orders introduced by Plesken and Zassenhaus. These are classified by the polytrope region. We advance the ideal theory of graduated orders by introducing their ideal class polytropes. This article emphasizes examples and computations. It offers first steps in the geometric combinatorics of endomorphism rings of configurations in affine buildings., Comment: 14 pages

Functionality of living cells is inherently linked to subunits with dimensions on the nanoscale. In case of osteoblasts the cell surface plays a particularly important role for adhesion and spreading which are crucial properties with regard to bone implants. Here we present a comprehensive characterization of the 3D nanomorphology of living as well as fixed osteoblastic cells using scanning ion conductance microscopy (SICM) which is a nanoprobing method largely avoiding forces. Dynamic ruffles are observed, manifesting themselves in characteristic membrane protrusions. They contribute to the overall surface corrugation which we systematically study by introducing the relative 3D excess area as a function of projected adhesion area. A clear anticorrelation is found upon analysis of ~40 different cells on glass as well as on amine covered surfaces. At the rim of lamellipodia characteristic edge heights between 100 nm and ~300 nm are observed. Power spectral densities of membrane fluctuations show frequency-dependent decay exponents in excess of -2 on living osteoblasts. We discuss the capability of apical membrane features and fluctuation dynamics in aiding assessment of adhesion and migration properties on a single-cell basis., Comment: 27 pages, 11 figures

We show that all elementary lattices that are free $\Z_p C_p$-modules admit an orthogonal decomposition into a sum of free unimodular and $p$-modular $\Z_p C_p$ lattices.

It is shown that extremal $2$-modular lattices of ranks $32$ and $48$ are generated by their vectors of minimal norm. In the proof, we use certain properties of the difference of normalized Hecke eigenforms. We refer to them as the pseudo-normalized Hecke eigenform, the concept of which is introduced in this paper., Comment: 15 pages

Building upon the application of flags to network coding introduced by Liebhold, Nebe, and Vazquez-Castro, we develop a variant of this coding technique that uses degenerate flags. The information set is a metric affine space isometric to the space of upper triangular matrices endowed with the flag rank metric. This suggests the development of a theory for flag rank metric codes in analogy to the rank metric codes used in linear subspace coding., Comment: Updated acknowledgements

Kaneko and Koike introduced the notion of extremal quasi-modular form and proposed conjectures on their arithmetic properties. The aim of this note is to prove a rather sharp multiplicity estimate for these quasi-modular forms. The note ends with discussions and partial answers around these conjectures and an appendix by G. Nebe containing the proof of the integrality of the Fourier coefficients of the normalised extremal quasimodular form of weight 14 and depth 1.

The methods to classify extremal unimodular lattices with given automorphisms are extended to the situation of modular lattices. A slightly more general notion than the type from the PhD thesis of Michael Juergens is the det-type. The det-type of an automorphism on $L$ determines the one of all partial dual lattices of $L$. This easy observation allows to exclude quite a few det-types of automorphisms left open in the above mentioned thesis. Passing to suitable p-maximal lattices, extremal l-modular lattices of composite level 14 and 15 of dimension 12 and the ones of level 6 and dimension 16 are classified.

It is shown that under some additional assumption two diagonalizable integral matrices X and Y with only rational eigenvalues are conjugate in GL(n,Z) if and only if they are conjugate over all localizations. This is used to prove that for a prime p == 3 (mod 4) the adjacency matrices of the Paley graph and the Peisert graph on p^2 vertices are conjugate in GL(p^2,Z), answering a question by Peter Sin.

We classify the dual strongly perfect lattices in dimension 16. There are four pairs of such lattices, the famous Barnes-Wall lattice $\Lambda _{16}$, the extremal 5-modular lattice $N_{16}$, the odd Barnes-Wall lattice $O_{16}$ and its dual, and one pair of new lattices $\Gamma _{16}$ and its dual. The latter pair belongs to a new infinite series of dual strongly perfect lattices, the sandwiched Barnes-Wall lattices, described by the authors in a previous paper. An updated table of all known strongly perfect lattices up to dimension 26 is available in the catalogue of lattices., Comment: 30 pages