Commutative algebraic monoid structures on affine surfaces

We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n - 1 n-1 or 𝑛 on a normal affine variety of dimension 𝑛.

Commutative algebraic monoid structures on affine spaces

We study commutative associative polynomial operations [Formula: see text] with unit on the affine space [Formula: see text] over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric varieties.

A Geometric Interpretation of the Intertwining Number

We exhibit a connection between two statistics on set partitions, the intertwining number and the depth-index. In particular, results link the intertwining number to the algebraic geometry of Borel orbits. Furthermore, by studying the generating polynomials of our statistics, we determine the $q=-1$ specialization of a $q$-analogue of the Bell numbers. Finally, by using Renner's $H$-polynomial of an algebraic monoid, we introduce and study a $t$-analog of $q$-Stirling numbers.

Affinely spanned quasi-stochastic algebraic monoids

A linear algebraic monoid over an algebraically closed field [Formula: see text] of characteristic zero is called (row) quasi-stochastic if each row of each matrix element is of sum one. Any linear algebraic monoid over [Formula: see text] can be embedded as an algebraic submonoid of the maximum affinely spanned quasi-stochastic monoid of some degree [Formula: see text]. The affinely spanned quasi-stochastic algebraic monoids form a basic class of quasi-stochastic algebraic monoids. An initial study of structure of affinely spanned quasi-stochastic algebraic monoids is conducted. Among other things, it is proved that the Zariski closure of a parabolic subgroup of the unit group of an affinely spanned quasi-stochastic algebraic monoid is affinely spanned.

THE BRAID ROOK MONOID

In linear algebraic monoid theory, the Renner monoids play the role of the Weyl groups in linear algebraic group theory. It is well known that Weyl groups are Coxeter groups, and that we can associate a Hecke algebra and an Artin–Tits group to each Coxeter group. The question of the existence of a Hecke algebra associated with each Renner monoid has been positively answered. In this paper we discuss the question of the existence of an equivalent of the Artin–Tits groups in the framework of Renner monoids. We consider the seminal case of the rook monoid and introduce a new length function.

Products of idempotents in algebraic monoids

Let M be a reductive algebraic monoid with zero and unit group G. We obtain a description of the submonoid generated by the idempotents of M. In particular, we find necessary and sufficient conditions for M\G to be idempotent generated.