Subalgebras generated in degree two with minimal Hilbert function
DOI:
https://doi.org/10.7146/math.scand.a-122603Abstract
What can be said about the subalgebras of the polynomial ring, with minimal or maximal Hilbert function? This question was discussed in a recent paper by M. Boij and A. Conca. In this paper we study the subalgebras generated in degree two with minimal Hilbert function. The problem to determine the generators of these algebras transfers into a combinatorial problem on counting maximal north-east lattice paths inside a shifted Ferrers diagram. We conjecture that the subalgebras generated in degree two with minimal Hilbert function are generated by an initial Lex or RevLex segment.
References
Boij, M. and Conca, A., On Fröberg-Macaulay conjectures for algebras, Rend. Istit. Mat. Univ. Trieste 50 (2018), 139–147. https://doi.org/10.13137/2464-8728/22433
Conca, A., Symmetric ladders, Nagoya Math. J. 136 (1994), 35–56. https://doi.org/10.1017/S0027763000024958
Corso, A. and Nagel, U., Specializations of Ferrers ideals, J. Algebraic Combin. 28 (2008), no. 3, 425–437. https://doi.org/10.1007/s10801-007-0111-2
Corso, A., Nagel, U., Petrović, S., and Yuen, C., Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas, Forum Math. 29 (2017), no. 4, 799–830. https://doi.org/10.1515/forum-2016-0007
Fröberg, R., An inequality for Hilbert series of graded algebras, Math. Scand. 56 (1985), no. 2, 117–144. https://doi.org/10.7146/math.scand.a-12092
Fröberg, R. and Lundqvist, S., Questions and conjectures on extremal Hilbert series, Rev. Un. Mat. Argentina 59 (2018), no. 2, 415–429. https://doi.org/10.33044/revuma.v59n2a10
Grayson, D. R. and Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, Available at https://faculty.math.illinois.edu/Macaulay2/.
Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL, 2018.
Conca, A., Symmetric ladders, Nagoya Math. J. 136 (1994), 35–56. https://doi.org/10.1017/S0027763000024958
Corso, A. and Nagel, U., Specializations of Ferrers ideals, J. Algebraic Combin. 28 (2008), no. 3, 425–437. https://doi.org/10.1007/s10801-007-0111-2
Corso, A., Nagel, U., Petrović, S., and Yuen, C., Blow-up algebras, determinantal ideals, and Dedekind-Mertens-like formulas, Forum Math. 29 (2017), no. 4, 799–830. https://doi.org/10.1515/forum-2016-0007
Fröberg, R., An inequality for Hilbert series of graded algebras, Math. Scand. 56 (1985), no. 2, 117–144. https://doi.org/10.7146/math.scand.a-12092
Fröberg, R. and Lundqvist, S., Questions and conjectures on extremal Hilbert series, Rev. Un. Mat. Argentina 59 (2018), no. 2, 415–429. https://doi.org/10.33044/revuma.v59n2a10
Grayson, D. R. and Stillman, M. E., Macaulay2, a software system for research in algebraic geometry, Available at https://faculty.math.illinois.edu/Macaulay2/.
Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL, 2018.
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Published
2021-02-17
How to Cite
Nicklasson, L. (2021). Subalgebras generated in degree two with minimal Hilbert function. MATHEMATICA SCANDINAVICA, 127(1), 5–27. https://doi.org/10.7146/math.scand.a-122603
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