The California State Teachers’ Retirement System (CalSTRS) is the nation’s second-largest public pension, with around $145 billion in assets under management. So when it speaks, companies tend to listen.

But I’m not so sure that Facebook will be among those willing to sit at E.F. Teacher’s knee.

CalSTRS reportedly is concerned with certain corporate governance provisions Facebook disclosed last week in its IPO filing, including those related to preemptive restrictions on proxy battles or hostile takeover attempts.

Here is how Janice Hester-Amey, a portfolio manager in CalSTRS Corporate Governance unit, explained it to Reuters: “No matter how brilliant you are, when you come to the public market — not that we want to ever tell Zuckerberg or anyone like him how to run his company — there should be some protection especially for long-term, patient money like CalSTRS.”

Hester-Amey added that CalSTRS has invested in Facebook via private equity funds and is expecting to invest in the social network’s publicly-traded shares.

If CalSTRS has any leverage with Facebook, it would have to be in that prospective purchase of publicly-traded shares, because its private equity investment is virtually nonexistent.

The pension system declined to discuss specifics (citing confidentiality), but Fortune has learned that its sole private equity exposure is through commitments to funds managed by Technology Crossover Ventures. The Palo Alto, Calif.-based firm did not invest in Facebook directly, but rather purchased shares on the secondary market from earlier backers like Accel Partners.

According to Bambi Francisco, TCV paid around $144 million to acquire Facebook common and preferred shares in 2010, via several transactions. My assumption is that the deals were done out of Technology Crossover Ventures VII, a $3 billion fund raised in late 2007. CalSTRS committed $200 million to that vehicle, which represents around a 6.67% stake. That means its total investment in Facebook works out to only around $9.6 million.

The value of those shares has obviously fluctuated in the intervening time, but not necessarily in the pension system’s favor. Francisco reported that TCV bought common shares at $45 a piece, compared to a recent secondary trading price of $40 per share. Even if Facebook hit a $100 billion valuation through IPO — the high end of an estimated $75 billion to $100 billion range — that still would put them at just about break-even. And its overall ownership percentage is three lengthy decimal points away from single digits.

[Update: A CalSTRS spokesman now says that the system had $30 million of exposure to Facebook as of June 30, 2011. He would not get more specific, but I remain certain that it’s only through TCV. Either the $30 million reflects a mark-up through Q2 of last year, or that TCV bought even more shares after Francisco’s article. Either way, $30 million represents just .03% of Facebook’s anticipated $100 billion valuation]

[Update II: A TCV spokesperson says the firm holds around a 1% stake in Facebook.]

Moreover, TCV is only mentioned once in Facebook’s entire IPO filing: A section in which TCV is listed among several existing shareholders would have “agreed to vote all of their shares as directed by, and granted an irrevocable proxy to, Mr. Zuckerberg at his discretion on all matters to be voted upon by stockholders.”

Did you catch the irony? CalSTRS is worried that Zuckerberg has too much power, at the expense of long-term shareholders like CalSTRS. But part of Zuckerberg’s control was facilitated by TCV, which is the firm through which CalSTRS first invested (and, theoretically, should hold more effective sway). That’s right: CalSTRS, by virtue of being a limited partner with TCV, was technically a party to the capitulation.

Look, it’s entirely possible that Facebook will hear CalSTRS out, because doing so is a quick and easy way to appease corporate governance advocates. But it won’t be pressured into making any changes, because CalSTRS doesn’t have enough shares to matter.

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