BIBLIOGRAPHY TO ACCOMPANY “SHIFTED SIMPLICIAL COMPLEXES AND ALGEBRAIC SHIFTING”

ART M. DUVAL

Abstract. This is a bibliography to accompany the slides from my talk at the AMS Regional Meeting at Binghamton University in October, 2003.

In general, a reference in the slides with a name and date has an obvious unique corresponding entry in the bibliography. Exceptions, and a few further explanations, are given below, organized by slide number.

Slide 6: One place to find an exposition of the idea that shifted complexes are “iterated near-cones”is [DRo00].

Slide 7: The basics of non-pure shellability are in [BW96], but also see [BW97], for instance for the canonical shelling of a shifted complex.

Slide 8: Exterior algebraic shifting goes back to [Ka84], though the best expositions may be [BK88], where I first read of it, or [Ka02], which contains (or at least seems to contain) everything that Gil Kalai knows or suspects to be true about algebraic shifting, including some brief historical notes. An earlier version of [Ka02] is [Ka93].

Slide 9: Herzog’s survey [He02] is probably a good place to start with symmetric shifting, including the cases where the ideal I is not squarefree monomial, i.e., does not come from a simplicial complex. The three papers [AH00, AHH00a, AHH00b] represent just the (combinatorial) tip of the iceberg of work on generic initial ideals with the revlex order.

Slide 10: Kalai [Ka02] lists more “axioms” [Ka02, Theorems 2.1 and 2.2], and a more extensive example [Ka02, Example 2.4] of how to use them to figure out the algebraic shift of a complex. That “axiom” 5 follows from the four previous “axioms” I learned from a talk by Isabella Novik.

Slide 11: The “earlier version” due to Kalai is from [Ka93], and may be found in [Ka02, Section 4.3].

Slide 21: Kook and Reiner made their conjecture based on one example in a coffeehouse. I learned of it through personal communication. The theorem at the bottom of this slide, as well as the theorem at the bottom of slide 22, is from [Du03].

Slides 23 and 24: Theorems about integrality are from [DRe02], and theorems about “spectrality” are from [Du03].

References

[AH00] A. Aramova, J. Herzog, “Almost regular sequences and Betti numbers”, Amer. J. Math. 122 (2000), 689–719.

[AHH00a] A. Aramova, J. Herzog, T. Hibi, “Ideals with stable Betti numbers”, Adv. Math. 152 (2000), 72–77.

[AHH00b] A. Aramova, J. Herzog, T. Hibi, “Shifting operations and graded Betti numbers”, J. Alg. Combin. 12 (2000), 207–222.

[BNT02] E. Babson, I. Novik, R. Thomas, “Symmetric iterated Betti numbers”, preprint, 2002.

arXiv:math.CO/0206063

[BCP99] D. Bayer, H. Charalambous, and S. Popescu, “Extremal Betti numbers and applications to monomial ideals”, J. Algebra 221 (1999), 497–512.

[BK88] A. Björner and G. Kalai, “An extended Euler-Poincaré theorem”, Acta Math. 161 (1988), 279–303.

[BW96] A. Björner and M. L. Wachs, “Shellable nonpure complexes and posets. I”, Trans. Amer. Math. Soc. 348 (1996), 1299–1327.

[BW97] A. Björner and M. L. Wachs, “Shellable nonpure complexes and posets. II”, Trans. Amer. Math. Soc. 349 (1997), 3945–3975.

[DW02] X. Dong and M. L. Wachs, “Combinatorial Laplacian of the matching complex”, Electron. J. Combin. 9 (2002), #R17, 11 pp.

[Du03] A. M. Duval, “A common recursion for Laplacians of matroids and shifted simplicial complexes”, preprint, 2003. arXiv:math.CO/0310327

[DRe02] A. M. Duval and V. Reiner, “Shifted simplicial complexes are Laplacian integral”, Trans. Amer. Math. Soc., 354 (2002), 4313–4344.

[DRo00] A. M. Duval and L. L. Rose, “Iterated homology of simplicial complexes”, J. Alg. Comb. 12 (2000), 279–294.

[DZ01] A. M. Duval and P. Zhang, “Iterated homology and decompositions of simplicial complexes”, Israel J. Math. 121 (2001), 313–331.

[FH98] J. Friedman and P. Hanlon, “On the Betti numbers of chessboard complexes”, J. Alg. Comb. 8 (1998), 193–203.

[Ga79] A. M. Garsia, “Combinatorial methods in the theory of Cohen-Macaulay rings”, Adv. Math. 38 (1980), 229–266.

[He02] J. Herzog, “Generic initial ideals and graded Betti numbers”, in Computational commutative algebra and combinatorics (Osaka, 1999), pp. 75–120, Adv. Stud. Pure Math., vol. 33, Math. Soc. Japan, Tokyo, 2002.

[Ka84] G. Kalai, “Characterization of f-vectors of families of convex sets in R^{d}. I. Necessity of Eckhoff’s conditions”, Israel J. Math. 48 (1984), 175–195.

[Ka91] G. Kalai, “The diameter of graphs of convex polytopes and f-vector theory”, in Applied geometry and discrete mathematics, pp. 387–411, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, Amer. Math. Soc., Providence, RI, 1991.

[Ka93] G. Kalai, “Algebraic Shifting”, unpublished manuscript (July 1993 version); updated and polished in [Ka02].

[Ka02] G. Kalai, “Algebraic shifting”, in Computational commutative algebra and combinatorics (Osaka, 1999), pp. 121–163, Adv. Stud. Pure Math., vol. 33, Math. Soc. Japan, Tokyo, 2002.

[Ko99] W. Kook, “Recursions in spectrum polynomial of matroids”, preprint, 1999.

http://hypatia.math.uri.edu/~andrewk/abstracts/

[KRS00] W. Kook, V. Reiner, and D. Stanton, “Combinatorial Laplacians of matroid complexes”, J. Amer. Math. Soc. 13 (2000), 129–148.

[St79] R. P. Stanley, “Balanced Cohen-Macaulay complexes,” Trans. Amer. Math. Soc. 249

(1979), 139–157.

E-mail address: artduval@math.utep.edu

Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX 79968-0514