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June  2020, 28(2): 951-959. doi: 10.3934/era.2020050

On the mod p Steenrod algebra and the Leibniz-Hopf algebra

Faculty of Arts and Sciences, Department of Mathematics, Eastern Mediterranean University, Famagusta, North Cyprus via Mersin 10, Turkey

* Corresponding author: neset.turgay@emu.edu.tr

Received  March 2020 Revised  May 2020 Published  June 2020

Let $ p $ be a fixed odd prime. The Bockstein free part of the mod $ p $ Steenrod algebra, $ \mathcal{A}_p $, can be defined as the quotient of the mod $ p $ reduction of the Leibniz Hopf algebra, $ \mathcal{F}_p $. We study the Hopf algebra epimorphism $ \pi\colon \mathcal{F}_p\to \mathcal{A}_p $ to investigate the canonical Hopf algebra conjugation in $ \mathcal{A}_p $ together with the conjugation operation in $ \mathcal{F}_p $. We also give a result about conjugation invariants in the mod 2 dual Leibniz Hopf algebra using its multiplicative algebra structure.

Citation: Neşet Deniz Turgay. On the mod p Steenrod algebra and the Leibniz-Hopf algebra. Electronic Research Archive, 2020, 28 (2) : 951-959. doi: 10.3934/era.2020050
References:
[1]

J. F. Adams, Lectures on generalised cohomology, in Category Theory, Homology Theory and their Applications, III, Lecture Notes in Mathematics, 99, Springer, Berlin, 1969, 1–138. doi: 10.1007/BFb0081960.  Google Scholar

[2]

D. Arnon, Monomial bases in the Steenrod algebra, J. Pure Appl. Algebra, 96 (1994), 215-223.  doi: 10.1016/0022-4049(94)90099-X.  Google Scholar

[3]

M. G. Barratt and H. R. Miller, On the anti-automorphism of the Steenrod algebra, in Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981), Contemp. Math., 12, Amer. Math. Soc., Providence, RI, 1982, 47–52.  Google Scholar

[4]

D. Bulacu, S. Caenepeel, F. Panaite and F. Van Oystaeyen, Quasi-Hopf Algebras, Encyclopedia of Mathematics and its Applications, 171, Cambridge University Press, Cambridge, 2019. doi: 10.1017/9781108582780.  Google Scholar

[5]

S. R. Bullet and I. G. Macdonald, On the Adem relations, Topology, 21 (1982), 329-332.  doi: 10.1016/0040-9383(82)90015-5.  Google Scholar

[6]

D. P. CarlisleG. Walker and R. M. W. Wood, The intersection of the admissible basis and the Milnor basis of the Steenrod algebra, J. Pure Appl. Algebra, 128 (1998), 1-10.  doi: 10.1016/S0022-4049(97)00035-2.  Google Scholar

[7]

M. D. Crossley, The Steenrod algebra and other copolynomial Hopf algebras, Bull. London Math. Soc., 32 (2000), 609-614.  doi: 10.1112/S0024609300007128.  Google Scholar

[8]

M. D. Crossley, Some Hopf algebras of words, Glasg. Math. J., 48 (2006), 575-582.  doi: 10.1017/S0017089506003302.  Google Scholar

[9]

M. Crossley and N. D. Turgay, Conjugation invariants in the mod 2 dual Leibniz-Hopf algebra, Comm. Algebra, 41 (2013), 3261-3266.  doi: 10.1080/00927872.2012.682675.  Google Scholar

[10]

M. Crossley and N. D. Turgay, Conjugation invariants in the Leibniz-Hopf algebra, J. Pure Appl. Algebra, 217 (2013), 2247-2254.  doi: 10.1016/j.jpaa.2013.03.003.  Google Scholar

[11]

M. D. Crossley and S. Whitehouse, On conjugation invariants in the dual Steenrod algebra, Proc. Amer. Math. Soc., 128 (2000), 2809-2818.  doi: 10.1090/S0002-9939-00-05283-7.  Google Scholar

[12]

M. D. Crossley and S. Whitehouse, Higher conjugation cohomology in commutative Hopf algebras, Proc. Edinb. Math. Soc. (2), 44 (2001), 19-26.  doi: 10.1017/S0013091599000826.  Google Scholar

[13]

D. M. Davis, The antiautomorphism of the Steenrod algebra, Proc. Amer. Math. Soc., 44 (1974), 235-236.  doi: 10.1090/S0002-9939-1974-0328934-1.  Google Scholar

[14]

V. G. Drinfel'd, Quasi-Hopf algebras, Leningr. Math. J., 1 (1990), 1419-1457.   Google Scholar

[15]

R. Ehrenborg, On posets and Hopf algebras, Adv. Math., 119 (1996), 1-25.  doi: 10.1006/aima.1996.0026.  Google Scholar

[16]

D. Y. Emelyanov and T. Y. Popelensky, On monomial bases in the mod $p$ Steenrod algebra, J. Fixed Point Theory Appl., 17 (2015), 341-353.  doi: 10.1007/s11784-014-0166-3.  Google Scholar

[17]

K. Emir, Graphical calculus of Hopf crossed modules, Hacettepe J. Math. Statistics, 49 (2020), 695-707.  doi: 10.15672/hujms.467966.  Google Scholar

[18]

M. Hazewinkel, Generalized overlapping shuffle algebras, J. Math. Sci. (New York), 106 (2001), 3168-3186.  doi: 10.1023/A:1011386821910.  Google Scholar

[19]

M. Hazewinkel, The algebra of quasi-symmetric functions is free over the integers, Adv. Math., 164 (2001), 283-300.  doi: 10.1006/aima.2001.2017.  Google Scholar

[20]

M. Hazewinkel, Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions. Monodromy and differential equations, Acta Appl. Math., 75 (2003), 55-83.  doi: 10.1023/A:1022323609001.  Google Scholar

[21]

M. Hazewinkel, Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions. II, Acta Appl. Math., 85 (2005), 319-340.  doi: 10.1007/s10440-004-5635-z.  Google Scholar

[22]

M. Hazewinkel, Explicit polynomial generators for the ring of quasisymmetric functions over the integers, Acta. Appl. Math., 109 (2010), 39-44.  doi: 10.1007/s10440-009-9439-z.  Google Scholar

[23]

S. Kaji, A Maple Code for the Dual Leibniz–Hopf Algebra. Available from: http://www.skaji.org/files/Leibniz-Hopf.mw. Google Scholar

[24]

I. Karaca and I. Y. Karaca, On conjugation in the mod-$p$ Steenrod algebra, Turkish J. Math., 24 (2000), 359-365.   Google Scholar

[25]

I. Karaca, Monomial bases in the mod-$p$ Steenrod algebra, Czechoslovak Math. J., 55 (2005), 699-707.  doi: 10.1007/s10587-005-0057-2.  Google Scholar

[26] S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511613104.  Google Scholar
[27]

C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177 (1995), 967-982.  doi: 10.1006/jabr.1995.1336.  Google Scholar

[28]

J. Milnor, The Steenrod algebra and its dual, Ann. of Math. (2), 67 (1958), 150-171.  doi: 10.2307/1969932.  Google Scholar

[29]

J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2), 81, (1965), 211–264. doi: 10.2307/1970615.  Google Scholar

[30]

K. G. Monks, Change of basis, monomial relations, and the $P_t^{s}$ bases for the Steenrod algebra, J. Pure Appl. Algebra, 125 (1998), 235-260.  doi: 10.1016/S0022-4049(96)00140-5.  Google Scholar

[31]

K. G. Monks, STEENROD: A Maple package for computing with the Steenrod algebra, 1995. Google Scholar

[32]

J.-P. Serre, Cohomologie modulo 2 des complexes d'Eilenberg-MacLane, Comment. Math. Helv., 27 (1953), 198-232.  doi: 10.1007/BF02564562.  Google Scholar

[33]

J. H. Silverman, Conjugation and excess in the Steenrod algebra, Proc. Amer. Math. Soc., 119 (1993), 657-661.  doi: 10.1090/S0002-9939-1993-1152292-8.  Google Scholar

[34]

N. E. Steenrod, Cohomology Operations, Annals of Math Studies, 50, Princeton University Press, Princeton, NJ, 1962.  Google Scholar

[35]

W. Stein, et al., Sage Mathematics Software (Version 5.4.1), The Sage Development Team, 2012. Available from: http://www.sagemath.org. Google Scholar

[36]

P. D. Straffin Jr., Identities for conjugation in the Steenrod algebra, Proc. Amer. Math. Soc., 49 (1975), 253-255.  doi: 10.1090/S0002-9939-1975-0380796-3.  Google Scholar

[37]

R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv., 28 (1954), 17-86.  doi: 10.1007/BF02566923.  Google Scholar

[38]

N. D. Turgay, On the conjugation invariant problem in the mod $p$ dual Steenrod algebra, Ital. J. Pure Appl. Math., 34 (2015), 151-158.   Google Scholar

[39]

N. D. Turgay, A remark on the conjugation in the Steenrod algebra, Commun. Korean Math. Soc., 30 (2015), 269-276.  doi: 10.4134/CKMS.2015.30.3.269.  Google Scholar

[40]

N. D. Turgay, An alternative approach to the Adem relations in the mod 2 Steenrod algebra, Turkish J. Mathematics, 38 (2014), 924-934.  doi: 10.3906/mat-1309-6.  Google Scholar

[41]

N. D. Turgay, An alternative approach to the Adem relations in the mod $p$ Steenrod algebra, C. R. Acad. Bulgare Sci., 70 (2017), 457-466.   Google Scholar

[42]

N. D. Turgay, Invariants under decomposition of the conjugation in the mod 2 dual Leibniz-Hopf algebra, Miskolc Math. Notes, 19 (2018), 1217-1222.  doi: 10.18514/MMN.2018.2591.  Google Scholar

[43]

N. D. Turgay and S. Kaji, The mod 2 dual Steenrod algebra as a subalgebra of the mod 2 dual Leibniz-Hopf algebra, J. Homotopy Relat. Struct., 12 (2017), 727-739.  doi: 10.1007/s40062-016-0163-x.  Google Scholar

[44]

N. D. Turgay and I. Karaca, The Arnon bases in the Steenrod algebra, Georgian Math. J., (2018). doi: 10.1515/gmj-2018-0076.  Google Scholar

[45]

G. Walker and R. M. W. Wood, The nilpotence height of ${S}q^{2n}$, Proc. Amer. Math. Soc., 124 (1996), 1291-1295.  doi: 10.1090/S0002-9939-96-03203-0.  Google Scholar

[46]

G. Walker and R. M. W. Wood, The nilpotence height of ${P}^{p^n}$, Math. Proc. Cambridge Philos. Soc., 123 (1998), 85-93.  doi: 10.1017/S0305004197001813.  Google Scholar

[47]

C. T. C. Wall, Generators and relations for the Steenrod algebra, Ann. of Math. (2), 72 (1960), 429-444.  doi: 10.2307/1970225.  Google Scholar

[48]

R. M. W. Wood, A note on bases and relations in the Steenrod algebra, Bull. London Math. Soc., 27 (1995), 380-386.  doi: 10.1112/blms/27.4.380.  Google Scholar

show all references

References:
[1]

J. F. Adams, Lectures on generalised cohomology, in Category Theory, Homology Theory and their Applications, III, Lecture Notes in Mathematics, 99, Springer, Berlin, 1969, 1–138. doi: 10.1007/BFb0081960.  Google Scholar

[2]

D. Arnon, Monomial bases in the Steenrod algebra, J. Pure Appl. Algebra, 96 (1994), 215-223.  doi: 10.1016/0022-4049(94)90099-X.  Google Scholar

[3]

M. G. Barratt and H. R. Miller, On the anti-automorphism of the Steenrod algebra, in Symposium on Algebraic Topology in honor of José Adem (Oaxtepec, 1981), Contemp. Math., 12, Amer. Math. Soc., Providence, RI, 1982, 47–52.  Google Scholar

[4]

D. Bulacu, S. Caenepeel, F. Panaite and F. Van Oystaeyen, Quasi-Hopf Algebras, Encyclopedia of Mathematics and its Applications, 171, Cambridge University Press, Cambridge, 2019. doi: 10.1017/9781108582780.  Google Scholar

[5]

S. R. Bullet and I. G. Macdonald, On the Adem relations, Topology, 21 (1982), 329-332.  doi: 10.1016/0040-9383(82)90015-5.  Google Scholar

[6]

D. P. CarlisleG. Walker and R. M. W. Wood, The intersection of the admissible basis and the Milnor basis of the Steenrod algebra, J. Pure Appl. Algebra, 128 (1998), 1-10.  doi: 10.1016/S0022-4049(97)00035-2.  Google Scholar

[7]

M. D. Crossley, The Steenrod algebra and other copolynomial Hopf algebras, Bull. London Math. Soc., 32 (2000), 609-614.  doi: 10.1112/S0024609300007128.  Google Scholar

[8]

M. D. Crossley, Some Hopf algebras of words, Glasg. Math. J., 48 (2006), 575-582.  doi: 10.1017/S0017089506003302.  Google Scholar

[9]

M. Crossley and N. D. Turgay, Conjugation invariants in the mod 2 dual Leibniz-Hopf algebra, Comm. Algebra, 41 (2013), 3261-3266.  doi: 10.1080/00927872.2012.682675.  Google Scholar

[10]

M. Crossley and N. D. Turgay, Conjugation invariants in the Leibniz-Hopf algebra, J. Pure Appl. Algebra, 217 (2013), 2247-2254.  doi: 10.1016/j.jpaa.2013.03.003.  Google Scholar

[11]

M. D. Crossley and S. Whitehouse, On conjugation invariants in the dual Steenrod algebra, Proc. Amer. Math. Soc., 128 (2000), 2809-2818.  doi: 10.1090/S0002-9939-00-05283-7.  Google Scholar

[12]

M. D. Crossley and S. Whitehouse, Higher conjugation cohomology in commutative Hopf algebras, Proc. Edinb. Math. Soc. (2), 44 (2001), 19-26.  doi: 10.1017/S0013091599000826.  Google Scholar

[13]

D. M. Davis, The antiautomorphism of the Steenrod algebra, Proc. Amer. Math. Soc., 44 (1974), 235-236.  doi: 10.1090/S0002-9939-1974-0328934-1.  Google Scholar

[14]

V. G. Drinfel'd, Quasi-Hopf algebras, Leningr. Math. J., 1 (1990), 1419-1457.   Google Scholar

[15]

R. Ehrenborg, On posets and Hopf algebras, Adv. Math., 119 (1996), 1-25.  doi: 10.1006/aima.1996.0026.  Google Scholar

[16]

D. Y. Emelyanov and T. Y. Popelensky, On monomial bases in the mod $p$ Steenrod algebra, J. Fixed Point Theory Appl., 17 (2015), 341-353.  doi: 10.1007/s11784-014-0166-3.  Google Scholar

[17]

K. Emir, Graphical calculus of Hopf crossed modules, Hacettepe J. Math. Statistics, 49 (2020), 695-707.  doi: 10.15672/hujms.467966.  Google Scholar

[18]

M. Hazewinkel, Generalized overlapping shuffle algebras, J. Math. Sci. (New York), 106 (2001), 3168-3186.  doi: 10.1023/A:1011386821910.  Google Scholar

[19]

M. Hazewinkel, The algebra of quasi-symmetric functions is free over the integers, Adv. Math., 164 (2001), 283-300.  doi: 10.1006/aima.2001.2017.  Google Scholar

[20]

M. Hazewinkel, Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions. Monodromy and differential equations, Acta Appl. Math., 75 (2003), 55-83.  doi: 10.1023/A:1022323609001.  Google Scholar

[21]

M. Hazewinkel, Symmetric functions, noncommutative symmetric functions, and quasisymmetric functions. II, Acta Appl. Math., 85 (2005), 319-340.  doi: 10.1007/s10440-004-5635-z.  Google Scholar

[22]

M. Hazewinkel, Explicit polynomial generators for the ring of quasisymmetric functions over the integers, Acta. Appl. Math., 109 (2010), 39-44.  doi: 10.1007/s10440-009-9439-z.  Google Scholar

[23]

S. Kaji, A Maple Code for the Dual Leibniz–Hopf Algebra. Available from: http://www.skaji.org/files/Leibniz-Hopf.mw. Google Scholar

[24]

I. Karaca and I. Y. Karaca, On conjugation in the mod-$p$ Steenrod algebra, Turkish J. Math., 24 (2000), 359-365.   Google Scholar

[25]

I. Karaca, Monomial bases in the mod-$p$ Steenrod algebra, Czechoslovak Math. J., 55 (2005), 699-707.  doi: 10.1007/s10587-005-0057-2.  Google Scholar

[26] S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511613104.  Google Scholar
[27]

C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra, 177 (1995), 967-982.  doi: 10.1006/jabr.1995.1336.  Google Scholar

[28]

J. Milnor, The Steenrod algebra and its dual, Ann. of Math. (2), 67 (1958), 150-171.  doi: 10.2307/1969932.  Google Scholar

[29]

J. W. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2), 81, (1965), 211–264. doi: 10.2307/1970615.  Google Scholar

[30]

K. G. Monks, Change of basis, monomial relations, and the $P_t^{s}$ bases for the Steenrod algebra, J. Pure Appl. Algebra, 125 (1998), 235-260.  doi: 10.1016/S0022-4049(96)00140-5.  Google Scholar

[31]

K. G. Monks, STEENROD: A Maple package for computing with the Steenrod algebra, 1995. Google Scholar

[32]

J.-P. Serre, Cohomologie modulo 2 des complexes d'Eilenberg-MacLane, Comment. Math. Helv., 27 (1953), 198-232.  doi: 10.1007/BF02564562.  Google Scholar

[33]

J. H. Silverman, Conjugation and excess in the Steenrod algebra, Proc. Amer. Math. Soc., 119 (1993), 657-661.  doi: 10.1090/S0002-9939-1993-1152292-8.  Google Scholar

[34]

N. E. Steenrod, Cohomology Operations, Annals of Math Studies, 50, Princeton University Press, Princeton, NJ, 1962.  Google Scholar

[35]

W. Stein, et al., Sage Mathematics Software (Version 5.4.1), The Sage Development Team, 2012. Available from: http://www.sagemath.org. Google Scholar

[36]

P. D. Straffin Jr., Identities for conjugation in the Steenrod algebra, Proc. Amer. Math. Soc., 49 (1975), 253-255.  doi: 10.1090/S0002-9939-1975-0380796-3.  Google Scholar

[37]

R. Thom, Quelques propriétés globales des variétés différentiables, Comment. Math. Helv., 28 (1954), 17-86.  doi: 10.1007/BF02566923.  Google Scholar

[38]

N. D. Turgay, On the conjugation invariant problem in the mod $p$ dual Steenrod algebra, Ital. J. Pure Appl. Math., 34 (2015), 151-158.   Google Scholar

[39]

N. D. Turgay, A remark on the conjugation in the Steenrod algebra, Commun. Korean Math. Soc., 30 (2015), 269-276.  doi: 10.4134/CKMS.2015.30.3.269.  Google Scholar

[40]

N. D. Turgay, An alternative approach to the Adem relations in the mod 2 Steenrod algebra, Turkish J. Mathematics, 38 (2014), 924-934.  doi: 10.3906/mat-1309-6.  Google Scholar

[41]

N. D. Turgay, An alternative approach to the Adem relations in the mod $p$ Steenrod algebra, C. R. Acad. Bulgare Sci., 70 (2017), 457-466.   Google Scholar

[42]

N. D. Turgay, Invariants under decomposition of the conjugation in the mod 2 dual Leibniz-Hopf algebra, Miskolc Math. Notes, 19 (2018), 1217-1222.  doi: 10.18514/MMN.2018.2591.  Google Scholar

[43]

N. D. Turgay and S. Kaji, The mod 2 dual Steenrod algebra as a subalgebra of the mod 2 dual Leibniz-Hopf algebra, J. Homotopy Relat. Struct., 12 (2017), 727-739.  doi: 10.1007/s40062-016-0163-x.  Google Scholar

[44]

N. D. Turgay and I. Karaca, The Arnon bases in the Steenrod algebra, Georgian Math. J., (2018). doi: 10.1515/gmj-2018-0076.  Google Scholar

[45]

G. Walker and R. M. W. Wood, The nilpotence height of ${S}q^{2n}$, Proc. Amer. Math. Soc., 124 (1996), 1291-1295.  doi: 10.1090/S0002-9939-96-03203-0.  Google Scholar

[46]

G. Walker and R. M. W. Wood, The nilpotence height of ${P}^{p^n}$, Math. Proc. Cambridge Philos. Soc., 123 (1998), 85-93.  doi: 10.1017/S0305004197001813.  Google Scholar

[47]

C. T. C. Wall, Generators and relations for the Steenrod algebra, Ann. of Math. (2), 72 (1960), 429-444.  doi: 10.2307/1970225.  Google Scholar

[48]

R. M. W. Wood, A note on bases and relations in the Steenrod algebra, Bull. London Math. Soc., 27 (1995), 380-386.  doi: 10.1112/blms/27.4.380.  Google Scholar

Table 1.   
Calculations under conjugation operation
$\chi(\mathcal{P}^i)$ ${\cal A}_3$ ${\cal A}_5$ ${\cal A}_7$ ${\cal A}_{11}$
$\chi(\mathcal{P}^1)$ $2\mathcal{P}^1$ $4\mathcal{P}^1$ $6\mathcal{P}^1$ $10\mathcal{P}^1$
$\chi(\mathcal{P}^2)$ $\mathcal{P}^2$ $\mathcal{P}^2$ $\mathcal{P}^2$ $\mathcal{P}^2$
$\chi(\mathcal{P}^3)$ $2\mathcal{P}^3$ 4$\mathcal{P}^3$ $6\mathcal{P}^3$ $10\mathcal{P}^3$
$\chi(\mathcal{P}^4)$ $\mathcal{P}^{3,1}$ $\mathcal{P}^4$ $\mathcal{P}^4$ $\mathcal{P}^4$
$\chi(\mathcal{P}^5$) $\mathcal{P}^5$ + $2\mathcal{P}^{4,1}$ $4\mathcal{P}^5$ $6\mathcal{P}^5$ $10\mathcal{P}^5$
$\chi(\mathcal{P}^6$) $\mathcal{P}^6$+$\mathcal{P}^{5,1}$ $\mathcal{P}^{5,1}$ $\mathcal{P}^6$ $\mathcal{P}^6$
$\chi(\mathcal{P}^7)$ $2\mathcal{P}^{6,1}$ $\mathcal{P}^7+4\mathcal{P}^{6,1}$ $6\mathcal{P}^7$ $10\mathcal{P}^7$
$\chi(\mathcal{P}^8)$ $\mathcal{P}^{6,2}$ $3\mathcal{P}^8+\mathcal{P}^{7,1}$ $\mathcal{P}^{7,1}$ $\mathcal{P}^8$
$\chi(\mathcal{P}^9)$ $2\mathcal{P}^9$+$\mathcal{P}^{8,1}$+$2\mathcal{P}^{7,2}$ $3\mathcal{P}^9+4\mathcal{P}^{8,1}$ $\mathcal{P}^9+6\mathcal{P}^{8,1}$ $10\mathcal{P}^9$
$\chi(\mathcal{P}^{10})$ $\mathcal{P}^{9,1}$+ $\mathcal{P}^{8,2}$ $\mathcal{P}^{10}+\mathcal{P}^{9,1}$ $5\mathcal{P}^{10}+\mathcal{P}^{9,1}$ $\mathcal{P}^{10}$
Calculations under conjugation operation
$\chi(\mathcal{P}^i)$ ${\cal A}_3$ ${\cal A}_5$ ${\cal A}_7$ ${\cal A}_{11}$
$\chi(\mathcal{P}^1)$ $2\mathcal{P}^1$ $4\mathcal{P}^1$ $6\mathcal{P}^1$ $10\mathcal{P}^1$
$\chi(\mathcal{P}^2)$ $\mathcal{P}^2$ $\mathcal{P}^2$ $\mathcal{P}^2$ $\mathcal{P}^2$
$\chi(\mathcal{P}^3)$ $2\mathcal{P}^3$ 4$\mathcal{P}^3$ $6\mathcal{P}^3$ $10\mathcal{P}^3$
$\chi(\mathcal{P}^4)$ $\mathcal{P}^{3,1}$ $\mathcal{P}^4$ $\mathcal{P}^4$ $\mathcal{P}^4$
$\chi(\mathcal{P}^5$) $\mathcal{P}^5$ + $2\mathcal{P}^{4,1}$ $4\mathcal{P}^5$ $6\mathcal{P}^5$ $10\mathcal{P}^5$
$\chi(\mathcal{P}^6$) $\mathcal{P}^6$+$\mathcal{P}^{5,1}$ $\mathcal{P}^{5,1}$ $\mathcal{P}^6$ $\mathcal{P}^6$
$\chi(\mathcal{P}^7)$ $2\mathcal{P}^{6,1}$ $\mathcal{P}^7+4\mathcal{P}^{6,1}$ $6\mathcal{P}^7$ $10\mathcal{P}^7$
$\chi(\mathcal{P}^8)$ $\mathcal{P}^{6,2}$ $3\mathcal{P}^8+\mathcal{P}^{7,1}$ $\mathcal{P}^{7,1}$ $\mathcal{P}^8$
$\chi(\mathcal{P}^9)$ $2\mathcal{P}^9$+$\mathcal{P}^{8,1}$+$2\mathcal{P}^{7,2}$ $3\mathcal{P}^9+4\mathcal{P}^{8,1}$ $\mathcal{P}^9+6\mathcal{P}^{8,1}$ $10\mathcal{P}^9$
$\chi(\mathcal{P}^{10})$ $\mathcal{P}^{9,1}$+ $\mathcal{P}^{8,2}$ $\mathcal{P}^{10}+\mathcal{P}^{9,1}$ $5\mathcal{P}^{10}+\mathcal{P}^{9,1}$ $\mathcal{P}^{10}$
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