Zoran Skoda Zbl-Lenagan

ZBL07767960 Launois, S.; Lenagan, T. H.; Nolan, B. M.

Total positivity is a quantum phenomenon: the Grassmannian case. Memoirs of the American Mathematical Society 1448.

(ISBN 978-1-4704-6694-7/pbk; 978-1-4704-7681-6/ebook)

Classification: 16-02 16T20 14M15 20G42 13P10 05E10

Keywords: quantum Grassmannian; totally nonnegative Grassmannian; torus-invariant prime ideals; positroid; quantum minor; Cauchon-Le diagram; partition subalgebra

Algebras representing quantum groups and their homogeneous spaces are among the most prominent examples in noncommutative geometry. To understand their representations and the underlying geometry, the prime and primitive spectra of such quantum algebras are studied from early 1990-s, starting with works of A. Joseph, T. J. Hodges, T. Levasseur, K. R. Goodearl and E. S. Getzler. Regarding that Grassmannians are projective varieties, analogous quantum homogeneous space of a quantum linear group, the quantum Grassmannian, is likewise represented by a graded associative algebra $\mathcal{O}_q(G_{mn}(\mathbb{F}))$ over a ground field $\mathbb{F}$, viewed as its homogeneous coordinate ring. Goodearl and Lenagan found that the prime ideals of $\mathcal{O}_q(G_{mn}(\mathbb{F}))$ are grouped into strata indexed by prime ideals invariant under a natural action of the algebraic torus $\mathcal{H}=(\mathbb{F}^\ast)^n$. Thus, a natural task, accomplished in [S. Launois, T. H. Lenagan, L. Rigal, Prime ideals in the quantum Grassmannian, Selecta Math. (N.S.) 13 (2008), no. 4, 697725] was to classify $\mathcal{H}$-invariant primes; they parametrized the $\mathcal{H}$-primes by a combinatorial object, Cauchon diagrams (of allowed size). The same combinatorial objects, under the name of Le diagrams appear in [A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math.CO/0609764, September 2006.] labelling stratification of another geometric object, totally nonnegative Grassmannian, into so called positroid cells, which by now found important applications outside algebra. The work under review goes beyond the bijection between $\mathcal{H}$-primes of $\mathcal{O}_q(G_{mn}(\mathbb{F}))$ and positroid cells, by showing that it is actually a homeomorphism between the poset of positroids under closure and the poset of $\mathcal{H}$-primes under inclusion. Second, they show how the Postnikov graph of a positroid cell determines a distinguished set of generators for the corresponding $\mathcal{H}$-prime as a 1-sided ideal in the ring of quantum matrices. The generating set consists of those quantum Plücker coordinates which belong to the $\mathcal{H}$-prime ideal in question. Quantum Plücker coordinates are quantum minors of size $m\times m$ in the quantum matrix ring $\mathcal{O}(M_{mn}(\mathbb{F}))$ (well studied homogeneous polynomials of order $m$ in standard generators $x_{ij}$ of $\mathcal{O}(M_{mn}(\mathbb{F}))$).

These results are obtained by a combination and precise adaptation of several specialized techniques in this subject. If $R$ is a commutative Noetherian domain and $\mathbb{F}$ its field of fractions, they consider the partition subalgebra $\mathcal{O}(Y_\lambda(R))$ in the quantum matrix ring $\mathcal{O}(M_{mn}(R))$ over $R$; this is the subalgebra generated by those $x_{ij}$ which fit into the Young diagram with $m$ rows, superimposed onto the quantum generic matrix and corresponding to a partition $\lambda$. Some known techniques for computation with quantum minors are generalized to $\mathcal{O}(Y_\lambda(R))$; appropriate choices of expressions for quantum minors typically have zeros at places where the generators are not in $\mathcal{O}(Y_\lambda(R))$. Quantum matrix rings and partition subalgebras are viewed as examples of a notion of a Quantum Nilpotent Algebra (QNA), an iterated Ore extensions of the ground field satisfying certain axioms involving also the deformation parameter $q\neq 0$ and an action of the algebraic torus. In [Effacement des dérivations et spectres premiers des algèbres quantiques (French, with English summary), J. Algebra 260 (2003), no. 2, 476–518], G. Cauchon developed so called deleting derivations algorithm for a QNA $A$ which is used to construct embedding of the prime spectrum of $A$ into prime spectra of simpler iterated Ore extension, and iterate the construction to finally embed into the prime spectrum of a much simpler algebra, the corresponding quantum affine space $\bar{A}$. The embedding of the prime spectra is $\mathcal{H}$-invariant, hence inducing the embedding of the $\mathcal{H}$-prime spectra. Of course, to have use of this fact, one has to characterize the image; Cauchon diagrams characterize the $\mathcal{H}$-prime ideals in quantum affine algebras which are in the image. This can in particular be applied to the partition subalgebras.

The connection between the partition subalgebras and $\mathcal{H}$-primes for quantum Grassmannian $\mathcal{O}(G_{mn}(\mathbb{F}))$ is coming from the noncommutative dehomogenization theory which provides an isomorphism $\Phi_\lambda$ between certain localization $S(\gamma)[\bar{\gamma}^{-1}]$ of the quantum Schubert variety $S(\gamma)$ associated to the quantum Plücker coordinate $\gamma$ and $\mathcal{O}_{q^{-1}}(Y_\lambda(\mathbb{F}))[Z^{\pm 1};\sigma]$. The Young tableau $Y_\lambda$ is determined by $\gamma$. The quantum Schubert variety is a $S(\gamma)$ quotient of $\mathcal{O}(G_{mn}(\mathbb{F}))$ and only one $\mathcal{H}$-prime survives in each $S(\gamma)[\bar{\gamma}^{-1}]$. Thus one can study this particular $\mathcal{H}$-prime using calculations in the corresponding partition subalgebra.

Further techniques are used: noncommutative Gröbner bases (in a variant developed in [K. Casteels, A graph theoretic method for determining generating sets of prime ideals in quantum matrices, J. Algebra 330 (2011) 188205]), the formalism of quantum graded algebras with a straightening law, matroids, and combinatorics of certain sequences of quantum Plücker coordinates called Grassmann necklaces to get closer description and relation to the positroid cells. It is shown that Grassmann necklaces generate so called separating Ore sets for $\mathcal{H}$-primes. Separating Ore sets are used to understand the Zariski topology on the prime spectrum, in particular to decide that a prime $Q$ in one stratum is not a subset of a prime $P$ in another stratum. The Ore condition for sets of powers of quantum Plücker coordinates is proved by a noncommutative dehomogenization of the result of the reviewer [Z. Škoda, Every quantum minor generates an Ore set, Int. Math. Res. Not. IMRN 16 (2008), Art. ID rnn063, 8] that every quantum minor generates an Ore set in the algebra of quantum matrices.

The article finishes with a proof of an earlier conjecture of Goodearl that the $\mathcal{H}$-prime ideals in quantum matrix algebras have polynormal sequences of generators (in fact, of quantum Plücker coordinates).

The memoir is readable and reasonably clean, having in mind specialized techniques. Authors also meticulously care about, and, when possible, minimize assumptions on the ground field and the deformation parameter.