We define a faithful linear monoidal functor from the partition category, and hence Deligne’s category $\mathrm{\underline{Re}p}(S\_t)$, to additive Karoubi envelope of Heisenberg category. show that induced map on Grothendieck rings is injective corresponds Kronecker coproduct symmetric functions.

Pinched Veronese rings are formed by removing an algebra generator from a subring of polynomial ring. We study the homological properties such rings, including Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco Martino classified Cohen-Macaulayness pinched maximum entry exponent vector monomial; we re-prove their results with semigroup methods correct omission small class e...

A module is called uniseriat if it has a unique composition series of finite length. A ring (always with 1) is called serial if its right and left free modules are direct sums of uniserial modules. Nakayama, who called these rings generalized uniserial rings, proved [21, Theorem 171 that every finitely generated module over a serial ring is a direct sum of uniserial modules. In section one we g...

Let k be a field. We show that locally presentable, k-linear categories $${\mathcal {C}}$$ dualizable in the sense identity functor can recovered as $$\coprod _i x_i\otimes f_i$$ for objects $$x_i\in {\mathcal and left adjoints $$f_i$$ from to $$\mathrm {Vect}_k$$ are products of copies . This partially confirms conjecture by Brandenburg, author T. Johnson-Freyd. Motivated this, we also charact...