Degeneration order of $3\times 3$ nilpotent matrix tuples
The degeneration order of simultaneous similarity classes of $3\times 3$ nilpotent matrix tuples is determined, and is shown to be given by rank conditions.
math.RT↗
arXiv subjects
Mátyás Domokos
Publications and source records attributed to Mátyás Domokos.
Separating versus ordinary Noether numbers
Let $G$ be a finite group and $K$ a field containing an element of multiplicative order $|G|$. It is shown that if $G$ has a cyclic subgroup of index at most $2$, then the separating Noether number over $K$ of $G$ coincides with the Noether number over $K$ of $G$. The same conclusion holds when $G$ is the direct product of a dihedral group and the $2$-element group. On the other hand, the smallest non-abelian groups $G$ are found for which the separating Noether number over $K$ is strictly less than the Noether number over $K$. Along the way the exact value of the separating Noether number is determined for all groups of order at most $16$. The results show in particular that unlike the ordinary Noether number, the separating Noether number of a non-abelian finite group may well be equal to the separating Noether number of a proper direct factor of the group.
math.AC↗
On the cones of classical groups
The cone of a classical group $G$ is an affine $G\times G$-variety. The aim of this note is to initiate its combinatorial study in the cases when $G$ is the complex orthogonal or symplectic group. The coordinate ring of the cone of $G$ is a finitely generated commutative graded algebra. First the $G\times G$-module structure of its homogeneous components is determined. This is used to compute the Hilbert series of this coordinate ring in the cases when $G$ is the orthogonal group $\mathrm{O}(3)$, $\mathrm{O}(4)$, the special orthogonal group $\mathrm{SO}(4)$, and when $G$ is the symplectic group $\mathrm{Sp}(4)$. It is concluded that the coordinate ring of the cone of $\mathrm{O}(3)$ is not Koszul, hence the vanishing ideal of this cone has no quadratic Gröbner basis (although it is minimally generated by quadratic elements).
math.RT↗
Separating polynomial invariants over non-closed fields of finite abelian groups
It is proved that for any finite dimensional representation of a prime order group over the field of rational numbers, polynomial invariants of degree at most $3$ separate the orbits. A result providing an upper degree bound for separating invariants for representations of finite abelian groups over algebraically closed base fields of non-modular characteristic is generalized for the case of base fields that are not algebraically closed (like the fields of real or rational numbers).
math.AC↗
The directed Cayley diameter and the Davenport constant
The directed Cayley diameter of a finite group is investigated in terms of the monoid of product-one sequences over the group, via the new notion of directed geodesic atoms. Two quantities associated to the set of directed geodesic atoms provide lower and upper bounds for the directed Cayley diameter. An algorithm for computing the directed geodesic atoms is implemented in GAP, and is applied to determine the above mentioned quantities for all non-abelian groups of order at most $42$, and for the alternating group of degree $5$. Furthermore, the small and large Davenport constants of all these groups are computed (excepting the large Davenport constant for $A_5$), extending thereby the formerly obtained results on the groups of order less than $32$. Along the way the directed Cayley diameter of a finite abelian group is expressed in terms of its invariants.
math.GR↗
Symmetric polynomials over finite fields
It is shown that two vectors with coordinates in the finite $q$-element field of characteristic $p$ belong to the same orbit under the natural action of the symmetric group if each of the elementary symmetric polynomials of degree $p^k,2p^k,\dots,(q-1)p^k$, $k=0,1,2,\dots$ has the same value on them. This separating set of polynomial invariants for the natural permutation representation of the symmetric group is not far from being minimal when $q=p$ and the dimension is large compared to $p$. A relatively small separating set of multisymmetric polynomials over the field of $q$ elements is derived.
math.AC↗
Low dimensional flow polytopes and their toric ideals
The toric ideal of a $d$-dimensional flow polytope has an initial ideal generated by square-free monomials of degree at most $d$. The toric ideal of a flow polytope of dimension at most four has an initial ideal generated by square-free monomials of degree at most two, with the only exception of the four-dimensional Birkhoff polytope, whose toric ideal has an initial ideal generated by a square-free cubic monomial. The proof is based on a method to classify certain compressed flow polytopes, and a construction of a quadratic pulling triangulation of them. Along the way compressed flow polytopes are classified up to dimension four, and their Ehrhart polynomials are computed.
math.AC↗
The Noether numbers and the Davenport constants of the groups of order less than 32
The computation of the Noether numbers of all groups of order less than thirty-two is completed. It turns out that for these groups in non-modular characteristic the Noether number is attained on a multiplicity free representation, it is strictly monotone on subgroups and factor groups, and it does not depend on the characteristic. Algorithms are developed and used to determine the small and large Davenport constants of these groups. For each of these groups the Noether number is greater than the small Davenport constant, whereas the first example of a group whose Noether number exceeds the large Davenport constant is found, answering partially a question posed by Geroldinger and Grynkiewicz.
math.GR↗
${\rm SL}(m,C)$-equivariant and translation covariant continuous tensor valuations
The space of continuous, ${\rm SL}(m,C)$-equivariant, $m\geq 2$, and translation covariant valuations taking values in the space of real symmetric tensors on $C^m\cong R^{2m}$ of rank $r\geq 0$ is completely described. The classification involves the moment tensor valuation for $r\geq 1$ and is analogous to the classification of the corresponding tensor valuations that are ${\rm SL}(2m,R)$-equivariant, although the method of proof cannot be adapted.
math.MG↗