AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Total Positivity is a Quantum Phenomenon: The Grassmannian Case
About this Title
S. Launois, T. H. Lenagan and B. M. Nolan
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 291, Number 1448
ISBNs: 978-1-4704-6694-7 (print); 978-1-4704-7681-6 (online)
DOI: https://doi.org/10.1090/memo/1448
Published electronically: November 8, 2023
Keywords: Quantum grassmannian,
totally nonnegative grassmannian,
torus-invariant prime ideals,
positroids,
quantum minors,
Cauchon-Le diagrams,
partition subalgebras
Table of Contents
Chapters
- 1. Introduction
- 2. Quantum matrices and partition subalgbras thereof
- 3. Quantum Nilpotent Algebras and their $\mathcal {H}$-primes
- 4. Cauchon-Le diagrams and Postnikov graphs
- 5. $\mathcal {H}$-primes in partition subalgebras: membership
- 6. Primes in $\mathcal {O}_q(M_{m,n}(R))$ from Cauchon-Le diagrams
- 7. $\mathcal {H}$-primes in partition subalgebras: generation
- 8. Primes in $\mathcal {O}_q(Y_\lambda (R))$ from Cauchon-Le diagrams
- 9. $\mathcal {H}$-primes in the quantum grassmannian: membership
- 10. $\mathcal {H}$-primes in ${\mathcal O}_q(G_{mn}(\mathbb {K}))$: generation
- 11. The link with the total nonnegative grassmannian and applications
Abstract
The main aim of this paper is to establish a deep link between the totally nonnegative grassmannian and the quantum grassmannian. More precisely, under the assumption that the deformation parameter $q$ is transcendental, we show that “quantum positroids” are completely prime ideals in the quantum grassmannian ${\mathcal O}_q(G_{mn}(\mathbb {F}))$. As a consequence, we obtain that torus-invariant prime ideals in the quantum grassmannian are generated by polynormal sequences of quantum Plücker coordinates and give a combinatorial description of these generating sets. We also give a topological description of the poset of torus-invariant prime ideals in ${\mathcal O}_q(G_{mn}(\mathbb {F}))$, and prove a version of the orbit method for torus-invariant objects. Finally, we construct separating Ore sets for all torus-invariant primes in ${\mathcal O}_q(G_{mn}(\mathbb {F}))$. The latter is the first step in the Brown-Goodearl strategy to establish the orbit method for (quantum) grassmannians.- Susama Agarwala and Siân Fryer, An algorithm to construct the Le diagram associated to a Grassmann necklace, Glasg. Math. J. 62 (2020), no. 1, 85–91. MR 4039001, DOI 10.1017/s001708951800054x
- Susama Agarwala and Eloi Marin-Amat, Wilson loop diagrams and positroids, Comm. Math. Phys. 350 (2017), no. 2, 569–601. MR 3607457, DOI 10.1007/s00220-016-2659-y
- Federico Ardila, Felipe Rincón, and Lauren Williams, Positroids and non-crossing partitions, Trans. Amer. Math. Soc. 368 (2016), no. 1, 337–363. MR 3413866, DOI 10.1090/tran/6331
- Nima Arkani-Hamed, Jacob Bourjaily, Freddy Cachazo, Alexander Goncharov, Alexander Postnikov, and Jaroslav Trnka, Grassmannian geometry of scattering amplitudes, Cambridge University Press, Cambridge, 2016. MR 3467729, DOI 10.1017/CBO9781316091548
- Jason P. Bell and Stéphane Launois, On the dimension of $H$-strata in quantum algebras, Algebra Number Theory 4 (2010), no. 2, 175–200. MR 2592018, DOI 10.2140/ant.2010.4.175
- J. Bell, S. Launois, and N. Nguyen, Dimension and enumeration of primitive ideals in quantum algebras, J. Algebraic Combin. 29 (2009), no. 3, 269–294. MR 2496308, DOI 10.1007/s10801-008-0132-5
- Michel Brion and Shrawan Kumar, Frobenius splitting methods in geometry and representation theory, Progress in Mathematics, vol. 231, Birkhäuser Boston, Inc., Boston, MA, 2005. MR 2107324
- Ken A. Brown and Ken R. Goodearl, Lectures on algebraic quantum groups, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2002. MR 1898492, DOI 10.1007/978-3-0348-8205-7
- Kenneth A. Brown and Kenneth R. Goodearl, Zariski topologies on stratified spectra of quantum algebras, Commutative algebra and noncommutative algebraic geometry. Vol. II, Math. Sci. Res. Inst. Publ., vol. 68, Cambridge Univ. Press, New York, 2015, pp. 63–91. MR 3496861
- K. A. Brown, K. R. Goodearl, and M. Yakimov, Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space, Adv. Math. 206 (2006), no. 2, 567–629. MR 2263715, DOI 10.1016/j.aim.2005.10.004
- José Bueso, José Gómez-Torrecillas, and Alain Verschoren, Algorithmic methods in non-commutative algebra, Mathematical Modelling: Theory and Applications, vol. 17, Kluwer Academic Publishers, Dordrecht, 2003. Applications to quantum groups. MR 2006329, DOI 10.1007/978-94-017-0285-0
- Karel Casteels, A graph theoretic method for determining generating sets of prime ideals in quantum matrices, J. Algebra 330 (2011), 188–205. MR 2774624, DOI 10.1016/j.jalgebra.2010.12.032
- Karel Casteels, Quantum matrices by paths, Algebra Number Theory 8 (2014), no. 8, 1857–1912. MR 3285618, DOI 10.2140/ant.2014.8.1857
- Karel Casteels and Siân Fryer, From Grassmann necklaces to restricted permutations and back again, Algebr. Represent. Theory 20 (2017), no. 4, 895–921. MR 3669163, DOI 10.1007/s10468-017-9668-1
- Gérard Cauchon, Effacement des dérivations et spectres premiers des algèbres quantiques, J. Algebra 260 (2003), no. 2, 476–518 (French, with English summary). MR 1967309, DOI 10.1016/S0021-8693(02)00542-2
- Gérard Cauchon, Spectre premier de $O_q(M_n(k))$: image canonique et séparation normale, J. Algebra 260 (2003), no. 2, 519–569 (French, with English summary). MR 1967310, DOI 10.1016/S0021-8693(02)00543-4
- Siân Fryer and Milen Yakimov, Separating Ore sets for prime ideals of quantum algebras, Bull. Lond. Math. Soc. 49 (2017), no. 2, 202–215. MR 3656289, DOI 10.1112/blms.12006
- Joel Geiger and Milen Yakimov, Quantum Schubert cells via representation theory and ring theory, Michigan Math. J. 63 (2014), no. 1, 125–157. MR 3189471, DOI 10.1307/mmj/1395234362
- K. R. Goodearl, S. Launois, and T. H. Lenagan, Totally nonnegative cells and matrix Poisson varieties, Adv. Math. 226 (2011), no. 1, 779–826. MR 2735775, DOI 10.1016/j.aim.2010.07.010
- K. R. Goodearl, S. Launois, and T. H. Lenagan, Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves, Math. Z. 269 (2011), no. 1-2, 29–45. MR 2836058, DOI 10.1007/s00209-010-0714-5
- K. R. Goodearl and T. H. Lenagan, Prime ideals invariant under winding automorphisms in quantum matrices, Internat. J. Math. 13 (2002), no. 5, 497–532. MR 1914562, DOI 10.1142/S0129167X02001393
- K. R. Goodearl and T. H. Lenagan, Winding-invariant prime ideals in quantum $3\times 3$ matrices, J. Algebra 260 (2003), no. 2, 657–687. MR 1967316, DOI 10.1016/S0021-8693(02)00566-5
- K. R. Goodearl and T. H. Lenagan, Quantized coinvariants at transcendental $q$, Hopf algebras in noncommutative geometry and physics, Lecture Notes in Pure and Appl. Math., vol. 239, Dekker, New York, 2005, pp. 155–165. MR 2106928
- K. R. Goodearl and E. S. Letzter, Prime ideals in skew and $q$-skew polynomial rings, Mem. Amer. Math. Soc. 109 (1994), no. 521, vi+106. MR 1197519, DOI 10.1090/memo/0521
- K. R. Goodearl and E. S. Letzter, The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras, Trans. Amer. Math. Soc. 352 (2000), no. 3, 1381–1403. MR 1615971, DOI 10.1090/S0002-9947-99-02345-4
- K. R. Goodearl and M. Yakimov, Poisson structures on affine spaces and flag varieties. II, Trans. Amer. Math. Soc. 361 (2009), no. 11, 5753–5780. MR 2529913, DOI 10.1090/S0002-9947-09-04654-6
- K. R. Goodearl and M. T. Yakimov, From quantum Ore extensions to quantum tori via noncommutative UFDs, Adv. Math. 300 (2016), 672–716. MR 3534843, DOI 10.1016/j.aim.2016.03.029
- Maria Gorelik, The prime and the primitive spectra of a quantum Bruhat cell translate, J. Algebra 227 (2000), no. 1, 211–253. MR 1754232, DOI 10.1006/jabr.1999.8235
- Xuhua He and Thomas Lam, Projected Richardson varieties and affine Schubert varieties, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 6, 2385–2412 (English, with English and French summaries). MR 3449584
- Christian Kassel, Quantum groups, Graduate Texts in Mathematics, vol. 155, Springer-Verlag, New York, 1995. MR 1321145, DOI 10.1007/978-1-4612-0783-2
- A. C. Kelly, T. H. Lenagan, and L. Rigal, Ring theoretic properties of quantum Grassmannians, J. Algebra Appl. 3 (2004), no. 1, 9–30. MR 2047633, DOI 10.1142/S0219498804000630
- Yuji Kodama and Lauren Williams, KP solitons and total positivity for the Grassmannian, Invent. Math. 198 (2014), no. 3, 637–699. MR 3279534, DOI 10.1007/s00222-014-0506-3
- Allen Knutson, Thomas Lam, and David E. Speyer, Projections of Richardson varieties, J. Reine Angew. Math. 687 (2014), 133–157. MR 3176610, DOI 10.1515/crelle-2012-0045
- Allen Knutson, Thomas Lam, and David E. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710–1752. MR 3123307, DOI 10.1112/S0010437X13007240
- Thomas Lam, Dimers, webs, and positroids, J. Lond. Math. Soc. (2) 92 (2015), no. 3, 633–656. MR 3431654, DOI 10.1112/jlms/jdv039
- Thomas Lam, Totally nonnegative Grassmannian and Grassmann polytopes, Current developments in mathematics 2014, Int. Press, Somerville, MA, 2016, pp. 51–152. MR 3468251
- S. Launois, Idéaux premiers $\mathcal {H}$-invariants de l’algèbre des matrices quantiques, PhD Thesis, Université de Reims Champagne-Ardenne, 2003.
- Stéphane Launois, Les idéaux premiers invariants de $O_q({\scr M}_{m,p}({\Bbb C}))$, J. Algebra 272 (2004), no. 1, 191–246 (French, with English summary). MR 2029032, DOI 10.1016/j.jalgebra.2003.05.005
- Stéphane Launois, Combinatorics of $\scr H$-primes in quantum matrices, J. Algebra 309 (2007), no. 1, 139–167. MR 2301235, DOI 10.1016/j.jalgebra.2006.10.023
- S. Launois, T. H. Lenagan, and L. Rigal, Quantum unique factorisation domains, J. London Math. Soc. (2) 74 (2006), no. 2, 321–340. MR 2269632, DOI 10.1112/S0024610706022927
- S. Launois, T. H. Lenagan, and L. Rigal, Prime ideals in the quantum Grassmannian, Selecta Math. (N.S.) 13 (2008), no. 4, 697–725. MR 2403308, DOI 10.1007/s00029-008-0054-z
- T. H. Lenagan and L. Rigal, Quantum graded algebras with a straightening law and the AS-Cohen-Macaulay property for quantum determinantal rings and quantum Grassmannians, J. Algebra 301 (2006), no. 2, 670–702. MR 2236763, DOI 10.1016/j.jalgebra.2005.10.021
- T. H. Lenagan and L. Rigal, Quantum analogues of Schubert varieties in the Grassmannian, Glasg. Math. J. 50 (2008), no. 1, 55–70. MR 2381732, DOI 10.1017/S0017089507003928
- T. H. Lenagan and E. J. Russell, Cyclic orders on the quantum Grassmannian, Arab. J. Sci. Eng. Sect. C Theme Issues 33 (2008), no. 2, 337–350 (English, with English and Arabic summaries). MR 2500045
- M. Movshev and A. Schwarz, Quantum deformation of planar amplitudes, J. High Energy Phys. 4 (2018), 121, front matter+19. MR 3801153, DOI 10.1007/jhep04(2018)121
- B. M. Nolan, A strong Dixmier-Moeglin equivalence for quantum Schubert cells and an open problem for quantum Plücker coordinates. PhD thesis, University of Kent, 2017.
- Suho Oh, Positroids and Schubert matroids, J. Combin. Theory Ser. A 118 (2011), no. 8, 2426–2435. MR 2834184, DOI 10.1016/j.jcta.2011.06.006
- A Postnikov, Total positivity, Grassmannians, and networks, arXiv:math.CO/0609764, September 2006.
- A. Ramanathan, Equations defining Schubert varieties and Frobenius splitting of diagonals, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 61–90. MR 908216
- K. Rietsch, Closure relations for totally nonnegative cells in $G/P$, Math. Res. Lett. 13 (2006), no. 5-6, 775–786. MR 2280774, DOI 10.4310/MRL.2006.v13.n5.a8
- Zoran Škoda, Every quantum minor generates an Ore set, Int. Math. Res. Not. IMRN 16 (2008), Art. ID rnn063, 8. MR 2435753, DOI 10.1093/imrn/rnn063
- Kelli Talaska, Combinatorial formulas for $\Gamma$-coordinates in a totally nonnegative Grassmannian, J. Combin. Theory Ser. A 118 (2011), no. 1, 58–66. MR 2737184, DOI 10.1016/j.jcta.2009.10.006
- Milen Yakimov, A classification of $H$-primes of quantum partial flag varieties, Proc. Amer. Math. Soc. 138 (2010), no. 4, 1249–1261. MR 2578519, DOI 10.1090/S0002-9939-09-10180-6
- Milen Yakimov, Invariant prime ideals in quantizations of nilpotent Lie algebras, Proc. Lond. Math. Soc. (3) 101 (2010), no. 2, 454–476. MR 2679698, DOI 10.1112/plms/pdq006
- Milen Yakimov, A proof of the Goodearl-Lenagan polynormality conjecture, Int. Math. Res. Not. IMRN 9 (2013), 2097–2132. MR 3053415, DOI 10.1093/imrn/rns111