We show that, in the ring of virtual species, \[ \mathcal{I}=\sum_{m\geq 0}(-1)^m\prod_{i=1}^{m}\bigl((E^{-1})^i-1\bigr), \] where $\mathcal{I}$ is the species of interval orders and $E^{-1}$ is the multiplicative inverse of the species $E$ of sets. The right-hand side is the virtual species of signed ballot matrices introduced by Claesson and Hannah. They showed that its signed cardinality counts labeled interval orders. We strengthen this to a species identity, which we prove twice: first algebraically and then bijectively, using a natural sign-reversing involution. The cycle index series of $\mathcal{I}$ specializes to the generating series for labeled and unlabeled interval orders. We describe the automorphism group of an interval order as a Young subgroup and prove the identity $\mathcal{I}=\mathcal{R}\circ E_+$, where $\mathcal{R}$ is the species of rigid interval orders. We also show that Glaisher's T-number $T_n$ counts the $24$-colored interval orders on $[n]$ in which no isolated element has color $24$.
For a simplicial complex $\Delta$ and integers $n_i\ge 2$, set $A_{\Delta,\mathbf n}=\mathbb{k}[x_1,\ldots,x_m]/(I_\Delta+(x_1^{n_1},\ldots,x_m^{n_m}))$. We give an exact combinatorial formula for the canonical trace for arbitrary truncation exponents and for an arbitrary simplicial complex after deleting irrelevant gh...
Let $\mathcal{F}^{m}=\{A\subset[m]\}$, endowed with the symmetric difference metric and the Cardinality Reverse-Lexicographic order. That way, we compute the homotopy type $\operatorname{\v{C}ech}\left(\mathcal{F}_{\preceq a}^m; \frac{r}{2}=1\right)$ by relating it to consecutively adding vertices of $\mathbb{I}^m$, th...
We extend the explicit octonionic approach to groups of type $\mathrm{E}_6$ to the twisted groups ${}^2\mathrm{SE}_{6,K}(F)$ associated with a quadratic Galois extension $K/F$. These groups act on the Albert space over $K$, preserving the Dickson--Freudenthal determinant and a Hermitean form. When the field norm $K^{\t...
J. Bray, Yegor Stepanov, Robert A. Wilson· 0 citations
Let $\nu_2(m)$ denote the $2$-adic valuation of a positive integer $m$ and set $N_m=m\bigl(1+\nu_2(m)/2\bigr)$. We prove four conjectures of Merca on the nonnegativity of truncated pentagonal number series weighted by the infinite products $\prod_{m\ge 1}(1-q^{2N_m})$ and $\prod_{r\ge 1}\bigl(q^{2^r r};q^{2^{r+1}r}\big...
Let $\boldsymbol{n}=(n_1,n_2,n_3):(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ be a holomorphic map germ admitting an injective representative. We prove that if $(d\boldsymbol{n})_0=0$, then $\mathrm{ord}_0(\boldsymbol{n})\in\{2,3,4\}$. This reduces L\^e's conjecture to excluding potential counterexamples $\boldsymbol{n}$ of o...
We show that von Neumann algebras of fundamental groups of closed orientable surfaces of genus $g\geq2$ are free group factors on $2g-1$generators. The key technical ingredient involves a proof that the element $w=ABA^{-1}B^{-1}$ of the free group $\mathbb{F}_{2}=\langle A,B\rangle$ is freely complemented in the group...
Dima Shlyakhtenko· 2 citations· ⚡1
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